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Theorem satfv1fvfmla1 34057
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 484 . . . . 5 ((𝑀𝑉𝐸𝑊) → 𝑀𝑉)
2 simpr 486 . . . . 5 ((𝑀𝑉𝐸𝑊) → 𝐸𝑊)
3 1onn 8591 . . . . . 6 1o ∈ ω
43a1i 11 . . . . 5 ((𝑀𝑉𝐸𝑊) → 1o ∈ ω)
51, 2, 43jca 1129 . . . 4 ((𝑀𝑉𝐸𝑊) → (𝑀𝑉𝐸𝑊 ∧ 1o ∈ ω))
653ad2ant1 1134 . . 3 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀𝑉𝐸𝑊 ∧ 1o ∈ ω))
7 satffun 34043 . . 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 1200 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐼 ∈ ω)
10 simp2r 1201 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐽 ∈ ω)
11 simp3l 1202 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐾 ∈ ω)
12 simp3r 1203 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝐿 ∈ ω)
13 satfv1fvfmla1.x . . . . . . . . . . 11 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))
14 eqid 2737 . . . . . . . . . . 11 {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))}
1513, 14pm3.2i 472 . . . . . . . . . 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 7369 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (𝑘𝑔𝑙) = (𝐾𝑔𝑙))
1817oveq2d 7378 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)))
1918eqeq2d 2748 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙))))
20 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐾 → (𝑎𝑘) = (𝑎𝐾))
2120breq1d 5120 . . . . . . . . . . . . . . 15 (𝑘 = 𝐾 → ((𝑎𝑘)𝐸(𝑎𝑙) ↔ (𝑎𝐾)𝐸(𝑎𝑙)))
2221notbid 318 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (¬ (𝑎𝑘)𝐸(𝑎𝑙) ↔ ¬ (𝑎𝐾)𝐸(𝑎𝑙)))
2322orbi2d 915 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → ((¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))))
2423rabbidv 3418 . . . . . . . . . . . 12 (𝑘 = 𝐾 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))})
2524eqeq2d 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 7370 . . . . . . . . . . . . 13 (𝑙 = 𝐿 → (𝐾𝑔𝑙) = (𝐾𝑔𝐿))
2827oveq2d 7378 . . . . . . . . . . . 12 (𝑙 = 𝐿 → ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿)))
2928eqeq2d 2748 . . . . . . . . . . 11 (𝑙 = 𝐿 → (𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))))
30 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐿 → (𝑎𝑙) = (𝑎𝐿))
3130breq2d 5122 . . . . . . . . . . . . . . 15 (𝑙 = 𝐿 → ((𝑎𝐾)𝐸(𝑎𝑙) ↔ (𝑎𝐾)𝐸(𝑎𝐿)))
3231notbid 318 . . . . . . . . . . . . . 14 (𝑙 = 𝐿 → (¬ (𝑎𝐾)𝐸(𝑎𝑙) ↔ ¬ (𝑎𝐾)𝐸(𝑎𝐿)))
3332orbi2d 915 . . . . . . . . . . . . 13 (𝑙 = 𝐿 → ((¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))))
3433rabbidv 3418 . . . . . . . . . . . 12 (𝑙 = 𝐿 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
3534eqeq2d 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 3595 . . . . . . . . 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 1372 . . . . . . . 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 872 . . . . . . 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 7369 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → (𝑖𝑔𝑗) = (𝐼𝑔𝑗))
4140oveq1d 7377 . . . . . . . . . . . 12 (𝑖 = 𝐼 → ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)))
4241eqeq2d 2748 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
43 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐼 → (𝑎𝑖) = (𝑎𝐼))
4443breq1d 5120 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝐼)𝐸(𝑎𝑗)))
4544notbid 318 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (¬ (𝑎𝑖)𝐸(𝑎𝑗) ↔ ¬ (𝑎𝐼)𝐸(𝑎𝑗)))
4645orbi1d 916 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ((¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))))
4746rabbidv 3418 . . . . . . . . . . . 12 (𝑖 = 𝐼 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})
4847eqeq2d 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 3214 . . . . . . . . 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 2738 . . . . . . . . . . . . 13 (𝑖 = 𝐼𝑛 = 𝑛)
5251, 40goaleq12d 33985 . . . . . . . . . . . 12 (𝑖 = 𝐼 → ∀𝑔𝑛(𝑖𝑔𝑗) = ∀𝑔𝑛(𝐼𝑔𝑗))
5352eqeq2d 2748 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗)))
54 eqeq1 2741 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (𝑖 = 𝑛𝐼 = 𝑛))
55 biidd 262 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)) ↔ if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗))))
5643breq1d 5120 . . . . . . . . . . . . . . . 16 (𝑖 = 𝐼 → ((𝑎𝑖)𝐸𝑧 ↔ (𝑎𝐼)𝐸𝑧))
5756, 44ifpbi23d 1081 . . . . . . . . . . . . . . 15 (𝑖 = 𝐼 → (if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)) ↔ if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗))))
5854, 55, 57ifpbi123d 1079 . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . 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 3418 . . . . . . . . . . . 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 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 3176 . . . . . . . . 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 918 . . . . . . . 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 7370 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → (𝐼𝑔𝑗) = (𝐼𝑔𝐽))
6665oveq1d 7377 . . . . . . . . . . . 12 (𝑗 = 𝐽 → ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙)))
6766eqeq2d 2748 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑋 = ((𝐼𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝑘𝑔𝑙))))
68 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑎𝑗) = (𝑎𝐽))
6968breq2d 5122 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → ((𝑎𝐼)𝐸(𝑎𝑗) ↔ (𝑎𝐼)𝐸(𝑎𝐽)))
7069notbid 318 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 → (¬ (𝑎𝐼)𝐸(𝑎𝑗) ↔ ¬ (𝑎𝐼)𝐸(𝑎𝐽)))
7170orbi1d 916 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → ((¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙)) ↔ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))))
7271rabbidv 3418 . . . . . . . . . . . 12 (𝑗 = 𝐽 → {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))} = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))})
7372eqeq2d 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 3214 . . . . . . . . 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 2738 . . . . . . . . . . . . 13 (𝑗 = 𝐽𝑛 = 𝑛)
7776, 65goaleq12d 33985 . . . . . . . . . . . 12 (𝑗 = 𝐽 → ∀𝑔𝑛(𝐼𝑔𝑗) = ∀𝑔𝑛(𝐼𝑔𝐽))
7877eqeq2d 2748 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝑋 = ∀𝑔𝑛(𝐼𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝐼𝑔𝐽)))
79 eqeq1 2741 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑗 = 𝑛𝐽 = 𝑛))
80 biidd 262 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑧𝐸𝑧𝑧𝐸𝑧))
8168breq2d 5122 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑧𝐸(𝑎𝑗) ↔ 𝑧𝐸(𝑎𝐽)))
8279, 80, 81ifpbi123d 1079 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)) ↔ if-(𝐽 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽))))
83 biidd 262 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → ((𝑎𝐼)𝐸𝑧 ↔ (𝑎𝐼)𝐸𝑧))
8479, 83, 69ifpbi123d 1079 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → (if-(𝑗 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝑗)) ↔ if-(𝐽 = 𝑛, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽))))
8582, 84ifpbi23d 1081 . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . 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 3418 . . . . . . . . . . . 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 2748 . . . . . . . . . . 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 632 . . . . . . . . . 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 3176 . . . . . . . . 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 918 . . . . . . . 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 3595 . . . . . . 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 1372 . . . . . 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 7396 . . . . . . . 8 𝑋 ∈ V
9594a1i 11 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ V)
96 ovex 7395 . . . . . . . 8 (𝑀m ω) ∈ V
9796rabex 5294 . . . . . . 7 {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))} ∈ V
98 eqeq1 2741 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑋 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
99 eqeq1 2741 . . . . . . . . . . . 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 3214 . . . . . . . . . 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 2741 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ↔ 𝑋 = ∀𝑔𝑛(𝑖𝑔𝑗)))
103 eqeq1 2741 . . . . . . . . . . . 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 3176 . . . . . . . . . 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 918 . . . . . . . . 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 3214 . . . . . . . 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 5495 . . . . . . 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 587 . . . . . 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 257 . . . . 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 873 . . . 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 4113 . . . 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 233 . . 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 2737 . . . . . 6 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
115114satfv1 33997 . . . . 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 2824 . . . 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 1134 . . 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 257 . 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 6899 . 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 205  wa 397  wo 846  if-wif 1062  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wrex 3074  {crab 3410  Vcvv 3448  cun 3913  c0 4287  cop 4597   class class class wbr 5110  {copab 5172  Fun wfun 6495  cfv 6501  (class class class)co 7362  ωcom 7807  1oc1o 8410  m cmap 8772  𝑔cgoe 33967  𝑔cgna 33968  𝑔cgol 33969   Sat csat 33970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-map 8774  df-goel 33974  df-gona 33975  df-goal 33976  df-sat 33977  df-fmla 33979
This theorem is referenced by:  satefvfmla1  34059
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