Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  satfv1fvfmla1 Structured version   Visualization version   GIF version

Theorem satfv1fvfmla1 32688
 Description: The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
Hypothesis
Ref Expression
satfv1fvfmla1.x 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
Assertion
Ref Expression
satfv1fvfmla1 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
Distinct variable groups:   𝐸,𝑎   𝐼,𝑎   𝐽,𝑎   𝐾,𝑎   𝐿,𝑎   𝑀,𝑎
Allowed substitution hints:   𝑉(𝑎)   𝑊(𝑎)   𝑋(𝑎)

Proof of Theorem satfv1fvfmla1
Dummy variables 𝑖 𝑗 𝑘 𝑛 𝑥 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . . 5 ((𝑀𝑉𝐸𝑊) → 𝑀𝑉)
2 simpr 488 . . . . 5 ((𝑀𝑉𝐸𝑊) → 𝐸𝑊)
3 1onn 8248 . . . . . 6 1o ∈ ω
43a1i 11 . . . . 5 ((𝑀𝑉𝐸𝑊) → 1o ∈ ω)
51, 2, 43jca 1125 . . . 4 ((𝑀𝑉𝐸𝑊) → (𝑀𝑉𝐸𝑊 ∧ 1o ∈ ω))
653ad2ant1 1130 . . 3 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉𝐸𝑊 ∧ 1o ∈ ω))
7 satffun 32674 . . 3 ((𝑀𝑉𝐸𝑊 ∧ 1o ∈ ω) → Fun ((𝑀 Sat 𝐸)‘1o))
86, 7syl 17 . 2 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → Fun ((𝑀 Sat 𝐸)‘1o))
9 simp2l 1196 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
10 simp2r 1197 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
11 simp3l 1198 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
12 simp3r 1199 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
13 satfv1fvfmla1.x . . . . . . . . . . 11 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
14 eqid 2824 . . . . . . . . . . 11 {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}
1513, 14pm3.2i 474 . . . . . . . . . 10 (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
1615a1i 11 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}))
17 oveq1 7145 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (𝑘𝑔𝑙) = (𝐾𝑔𝑙))
1817oveq2d 7154 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)))
1918eqeq2d 2835 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙))))
20 fveq2 6651 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐾 → (𝑎𝑘) = (𝑎𝐾))
2120breq1d 5057 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → ((𝑎𝑘)𝐸(𝑎𝑙) ↔ (𝑎𝐾)𝐸(𝑎𝑙)))
2221notbid 321 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (¬ (𝑎𝑘)𝐸(𝑎𝑙) ↔ ¬ (𝑎𝐾)𝐸(𝑎𝑙)))
2322orbi2d 913 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → ((¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))))
2423rabbidv 3465 . . . . . . . . . . . 12 (𝑘 = 𝐾 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))})
2524eqeq2d 2835 . . . . . . . . . . 11 (𝑘 = 𝐾 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))}))
2619, 25anbi12d 633 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))})))
27 oveq2 7146 . . . . . . . . . . . . 13 (𝑙 = 𝐿 → (𝐾𝑔𝑙) = (𝐾𝑔𝐿))
2827oveq2d 7154 . . . . . . . . . . . 12 (𝑙 = 𝐿 → ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)))
2928eqeq2d 2835 . . . . . . . . . . 11 (𝑙 = 𝐿 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))))
30 fveq2 6651 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐿 → (𝑎𝑙) = (𝑎𝐿))
3130breq2d 5059 . . . . . . . . . . . . . . 15 (𝑙 = 𝐿 → ((𝑎𝐾)𝐸(𝑎𝑙) ↔ (𝑎𝐾)𝐸(𝑎𝐿)))
3231notbid 321 . . . . . . . . . . . . . 14 (𝑙 = 𝐿 → (¬ (𝑎𝐾)𝐸(𝑎𝑙) ↔ ¬ (𝑎𝐾)𝐸(𝑎𝐿)))
3332orbi2d 913 . . . . . . . . . . . . 13 (𝑙 = 𝐿 → ((¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))))
3433rabbidv 3465 . . . . . . . . . . . 12 (𝑙 = 𝐿 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
3534eqeq2d 2835 . . . . . . . . . . 11 (𝑙 = 𝐿 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}))
3629, 35anbi12d 633 . . . . . . . . . 10 (𝑙 = 𝐿 → ((𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))}) ↔ (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})))
3726, 36rspc2ev 3620 . . . . . . . . 9 ((𝐾 ∈ ω ∧ 𝐿 ∈ ω ∧ (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}))
3811, 12, 16, 37syl3anc 1368 . . . . . . . 8 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}))
3938orcd 870 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})))
40 oveq1 7145 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → (𝑖𝑔𝑗) = (𝐼𝑔𝑗))
4140oveq1d 7153 . . . . . . . . . . . 12 (𝑖 = 𝐼 → ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)))
4241eqeq2d 2835 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
43 fveq2 6651 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐼 → (𝑎𝑖) = (𝑎𝐼))
4443breq1d 5057 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝐼)𝐸(𝑎𝑗)))
4544notbid 321 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (¬ (𝑎𝑖)𝐸(𝑎𝑗) ↔ ¬ (𝑎𝐼)𝐸(𝑎𝑗)))
4645orbi1d 914 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ((¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))))
4746rabbidv 3465 . . . . . . . . . . . 12 (𝑖 = 𝐼 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})
4847eqeq2d 2835 . . . . . . . . . . 11 (𝑖 = 𝐼 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}))
4942, 48anbi12d 633 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
50492rexbidv 3292 . . . . . . . . 9 (𝑖 = 𝐼 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
51 eqidd 2825 . . . . . . . . . . . . 13 (𝑖 = 𝐼𝑛 = 𝑛)
5251, 40goaleq12d 32616 . . . . . . . . . . . 12 (𝑖 = 𝐼 → ∀𝑔𝑛(𝑖𝑔𝑗) = ∀𝑔𝑛(𝐼𝑔𝑗))
5352eqeq2d 2835 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗)))
54 eqeq1 2828 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (𝑖 = 𝑛𝐼 = 𝑛))
55 biidd 265 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)) ↔ if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗))))
5643breq1d 5057 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐼 → ((𝑎𝑖)𝐸𝑧 ↔ (𝑎𝐼)𝐸𝑧))
5756, 44ifpbi23d 1077 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)) ↔ if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗))))
5854, 55, 57ifpbi123d 1075 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗))) ↔ if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))))
5958ralbidv 3191 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → (∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗))) ↔ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))))
6059rabbidv 3465 . . . . . . . . . . . 12 (𝑖 = 𝐼 → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))})
6160eqeq2d 2835 . . . . . . . . . . 11 (𝑖 = 𝐼 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))}))
6253, 61anbi12d 633 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}) ↔ (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))})))
6362rexbidv 3289 . . . . . . . . 9 (𝑖 = 𝐼 → (∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))})))
6450, 63orbi12d 916 . . . . . . . 8 (𝑖 = 𝐼 → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))}))))
65 oveq2 7146 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → (𝐼𝑔𝑗) = (𝐼𝑔𝐽))
6665oveq1d 7153 . . . . . . . . . . . 12 (𝑗 = 𝐽 → ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)))
6766eqeq2d 2835 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙))))
68 fveq2 6651 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑎𝑗) = (𝑎𝐽))
6968breq2d 5059 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → ((𝑎𝐼)𝐸(𝑎𝑗) ↔ (𝑎𝐼)𝐸(𝑎𝐽)))
7069notbid 321 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 → (¬ (𝑎𝐼)𝐸(𝑎𝑗) ↔ ¬ (𝑎𝐼)𝐸(𝑎𝐽)))
7170orbi1d 914 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → ((¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))))
7271rabbidv 3465 . . . . . . . . . . . 12 (𝑗 = 𝐽 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})
7372eqeq2d 2835 . . . . . . . . . . 11 (𝑗 = 𝐽 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}))
7467, 73anbi12d 633 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
75742rexbidv 3292 . . . . . . . . 9 (𝑗 = 𝐽 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
76 eqidd 2825 . . . . . . . . . . . . 13 (𝑗 = 𝐽𝑛 = 𝑛)
7776, 65goaleq12d 32616 . . . . . . . . . . . 12 (𝑗 = 𝐽 → ∀𝑔𝑛(𝐼𝑔𝑗) = ∀𝑔𝑛(𝐼𝑔𝐽))
7877eqeq2d 2835 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽)))
79 eqeq1 2828 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑗 = 𝑛𝐽 = 𝑛))
80 biidd 265 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑧𝐸𝑧𝑧𝐸𝑧))
8168breq2d 5059 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑧𝐸(𝑎𝑗) ↔ 𝑧𝐸(𝑎𝐽)))
8279, 80, 81ifpbi123d 1075 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)) ↔ if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
83 biidd 265 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → ((𝑎𝐼)𝐸𝑧 ↔ (𝑎𝐼)𝐸𝑧))
8479, 83, 69ifpbi123d 1075 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)) ↔ if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
8582, 84ifpbi23d 1077 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 → (if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗))) ↔ if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
8685ralbidv 3191 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → (∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗))) ↔ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))))
8786rabbidv 3465 . . . . . . . . . . . 12 (𝑗 = 𝐽 → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})
8887eqeq2d 2835 . . . . . . . . . . 11 (𝑗 = 𝐽 → ({𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))}))
8978, 88anbi12d 633 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))}) ↔ (𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})))
9089rexbidv 3289 . . . . . . . . 9 (𝑗 = 𝐽 → (∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})))
9175, 90orbi12d 916 . . . . . . . 8 (𝑗 = 𝐽 → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))}))))
9264, 91rspc2ev 3620 . . . . . . 7 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑛, if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))}))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})))
939, 10, 39, 92syl3anc 1368 . . . . . 6 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})))
9413ovexi 7172 . . . . . . . 8 𝑋 ∈ V
9594a1i 11 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ V)
96 ovex 7171 . . . . . . . 8 (𝑀m ω) ∈ V
9796rabex 5216 . . . . . . 7 {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} ∈ V
98 eqeq1 2828 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
99 eqeq1 2828 . . . . . . . . . . . 12 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}))
10098, 99bi2anan9 638 . . . . . . . . . . 11 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → ((𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
1011002rexbidv 3292 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})))
102 eqeq1 2828 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗)))
103 eqeq1 2828 . . . . . . . . . . . 12 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))} ↔ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))
104102, 103bi2anan9 638 . . . . . . . . . . 11 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → ((𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}) ↔ (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})))
105104rexbidv 3289 . . . . . . . . . 10 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → (∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}) ↔ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})))
106101, 105orbi12d 916 . . . . . . . . 9 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → ((∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))))
1071062rexbidv 3292 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))})) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))))
108107opelopabga 5401 . . . . . . 7 ((𝑋 ∈ V ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} ∈ V) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))))
10995, 97, 108sylancl 589 . . . . . 6 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))))
11093, 109mpbird 260 . . . . 5 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))})
111110olcd 871 . . . 4 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘∅) ∨ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
112 elun 4109 . . . 4 (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}) ↔ (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘∅) ∨ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
113111, 112sylibr 237 . . 3 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
114 eqid 2824 . . . . . 6 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
115114satfv1 32628 . . . . 5 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘1o) = (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
116115eleq2d 2901 . . . 4 ((𝑀𝑉𝐸𝑊) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1o) ↔ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))})))
1171163ad2ant1 1130 . . 3 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1o) ↔ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))})))
118113, 117mpbird 260 . 2 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1o))
119 funopfv 6698 . 2 (Fun ((𝑀 Sat 𝐸)‘1o) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}⟩ ∈ ((𝑀 Sat 𝐸)‘1o) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}))
1208, 118, 119sylc 65 1 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  if-wif 1058   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3132  ∃wrex 3133  {crab 3136  Vcvv 3479   ∪ cun 3916  ∅c0 4274  ⟨cop 4554   class class class wbr 5047  {copab 5109  Fun wfun 6330  ‘cfv 6336  (class class class)co 7138  ωcom 7563  1oc1o 8078   ↑m cmap 8389  ∈𝑔cgoe 32598  ⊼𝑔cgna 32599  ∀𝑔cgol 32600   Sat csat 32601 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-map 8391  df-goel 32605  df-gona 32606  df-goal 32607  df-sat 32608  df-fmla 32610 This theorem is referenced by:  satefvfmla1  32690
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