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Theorem fmla1 32783
 Description: The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎⊼𝑔𝑏) and ∀𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥 ∈ 𝑦. (Contributed by AV, 20-Sep-2023.)
Assertion
Ref Expression
fmla1 (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
Distinct variable group:   𝑖,𝑗,𝑥,𝑘,𝑙

Proof of Theorem fmla1
Dummy variables 𝑦 𝑛 𝑧 𝑚 𝑜 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-1o 8092 . . 3 1o = suc ∅
21fveq2i 6653 . 2 (Fmla‘1o) = (Fmla‘suc ∅)
3 peano1 7588 . . 3 ∅ ∈ ω
4 fmlasuc 32782 . . 3 (∅ ∈ ω → (Fmla‘suc ∅) = ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)}))
53, 4ax-mp 5 . 2 (Fmla‘suc ∅) = ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)})
6 fmla0xp 32779 . . 3 (Fmla‘∅) = ({∅} × (ω × ω))
7 fmla0 32778 . . . . . 6 (Fmla‘∅) = {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)}
87rexeqi 3363 . . . . 5 (∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
9 eqeq1 2802 . . . . . . . . . 10 (𝑦 = 𝑝 → (𝑦 = (𝑖𝑔𝑗) ↔ 𝑝 = (𝑖𝑔𝑗)))
1092rexbidv 3259 . . . . . . . . 9 (𝑦 = 𝑝 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗)))
1110elrab 3628 . . . . . . . 8 (𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} ↔ (𝑝 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗)))
12 oveq1 7147 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑖𝑔𝑗) → (𝑝𝑔𝑞) = ((𝑖𝑔𝑗)⊼𝑔𝑞))
1312eqeq2d 2809 . . . . . . . . . . . . . . 15 (𝑝 = (𝑖𝑔𝑗) → (𝑥 = (𝑝𝑔𝑞) ↔ 𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞)))
1413rexbidv 3256 . . . . . . . . . . . . . 14 (𝑝 = (𝑖𝑔𝑗) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ↔ ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞)))
15 eqidd 2799 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑖𝑔𝑗) → 𝑘 = 𝑘)
16 id 22 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑖𝑔𝑗) → 𝑝 = (𝑖𝑔𝑗))
1715, 16goaleq12d 32747 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑖𝑔𝑗) → ∀𝑔𝑘𝑝 = ∀𝑔𝑘(𝑖𝑔𝑗))
1817eqeq2d 2809 . . . . . . . . . . . . . . 15 (𝑝 = (𝑖𝑔𝑗) → (𝑥 = ∀𝑔𝑘𝑝𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
1918rexbidv 3256 . . . . . . . . . . . . . 14 (𝑝 = (𝑖𝑔𝑗) → (∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝 ↔ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
2014, 19orbi12d 916 . . . . . . . . . . . . 13 (𝑝 = (𝑖𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
21 eqeq1 2802 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑞 → (𝑧 = (𝑘𝑔𝑙) ↔ 𝑞 = (𝑘𝑔𝑙)))
22212rexbidv 3259 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑞 → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑧 = (𝑘𝑔𝑙) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘𝑔𝑙)))
23 fmla0 32778 . . . . . . . . . . . . . . . . . 18 (Fmla‘∅) = {𝑧 ∈ V ∣ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑧 = (𝑘𝑔𝑙)}
2422, 23elrab2 3631 . . . . . . . . . . . . . . . . 17 (𝑞 ∈ (Fmla‘∅) ↔ (𝑞 ∈ V ∧ ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘𝑔𝑙)))
25 oveq2 7148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑞 = (𝑘𝑔𝑙) → ((𝑖𝑔𝑗)⊼𝑔𝑞) = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)))
2625eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . 21 (𝑞 = (𝑘𝑔𝑙) → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) ↔ 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
2726biimpcd 252 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → (𝑞 = (𝑘𝑔𝑙) → 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
2827reximdv 3232 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → (∃𝑙 ∈ ω 𝑞 = (𝑘𝑔𝑙) → ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
2928reximdv 3232 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘𝑔𝑙) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
3029com12 32 . . . . . . . . . . . . . . . . 17 (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑞 = (𝑘𝑔𝑙) → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
3124, 30simplbiim 508 . . . . . . . . . . . . . . . 16 (𝑞 ∈ (Fmla‘∅) → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
3231rexlimiv 3239 . . . . . . . . . . . . . . 15 (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) → ∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)))
3332orim1i 907 . . . . . . . . . . . . . 14 ((∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) → (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
34 r19.43 3304 . . . . . . . . . . . . . 14 (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑘 ∈ ω ∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
3533, 34sylibr 237 . . . . . . . . . . . . 13 ((∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑖𝑔𝑗)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
3620, 35syl6bi 256 . . . . . . . . . . . 12 (𝑝 = (𝑖𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
3736com12 32 . . . . . . . . . . 11 ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (𝑝 = (𝑖𝑔𝑗) → ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
3837reximdv 3232 . . . . . . . . . 10 ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗) → ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
3938reximdv 3232 . . . . . . . . 9 ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4039com12 32 . . . . . . . 8 (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4111, 40simplbiim 508 . . . . . . 7 (𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))))
4241rexlimiv 3239 . . . . . 6 (∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
43 oveq1 7147 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
4443oveq1d 7155 . . . . . . . . . . . 12 (𝑖 = 𝑚 → ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)))
4544eqeq2d 2809 . . . . . . . . . . 11 (𝑖 = 𝑚 → (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
4645rexbidv 3256 . . . . . . . . . 10 (𝑖 = 𝑚 → (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙))))
47 eqidd 2799 . . . . . . . . . . . 12 (𝑖 = 𝑚𝑘 = 𝑘)
4847, 43goaleq12d 32747 . . . . . . . . . . 11 (𝑖 = 𝑚 → ∀𝑔𝑘(𝑖𝑔𝑗) = ∀𝑔𝑘(𝑚𝑔𝑗))
4948eqeq2d 2809 . . . . . . . . . 10 (𝑖 = 𝑚 → (𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗) ↔ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗)))
5046, 49orbi12d 916 . . . . . . . . 9 (𝑖 = 𝑚 → ((∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗))))
5150rexbidv 3256 . . . . . . . 8 (𝑖 = 𝑚 → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗))))
52 oveq2 7148 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
5352oveq1d 7155 . . . . . . . . . . . 12 (𝑗 = 𝑛 → ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)))
5453eqeq2d 2809 . . . . . . . . . . 11 (𝑗 = 𝑛 → (𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙))))
5554rexbidv 3256 . . . . . . . . . 10 (𝑗 = 𝑛 → (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙))))
56 eqidd 2799 . . . . . . . . . . . 12 (𝑗 = 𝑛𝑘 = 𝑘)
5756, 52goaleq12d 32747 . . . . . . . . . . 11 (𝑗 = 𝑛 → ∀𝑔𝑘(𝑚𝑔𝑗) = ∀𝑔𝑘(𝑚𝑔𝑛))
5857eqeq2d 2809 . . . . . . . . . 10 (𝑗 = 𝑛 → (𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗) ↔ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
5955, 58orbi12d 916 . . . . . . . . 9 (𝑗 = 𝑛 → ((∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))))
6059rexbidv 3256 . . . . . . . 8 (𝑗 = 𝑛 → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑗)) ↔ ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))))
6151, 60cbvrex2vw 3409 . . . . . . 7 (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) ↔ ∃𝑚 ∈ ω ∃𝑛 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
62 oveq1 7147 . . . . . . . . . . . . . 14 (𝑘 = 𝑜 → (𝑘𝑔𝑙) = (𝑜𝑔𝑙))
6362oveq2d 7156 . . . . . . . . . . . . 13 (𝑘 = 𝑜 → ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)))
6463eqeq2d 2809 . . . . . . . . . . . 12 (𝑘 = 𝑜 → (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ↔ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))))
6564rexbidv 3256 . . . . . . . . . . 11 (𝑘 = 𝑜 → (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ↔ ∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))))
66 id 22 . . . . . . . . . . . . 13 (𝑘 = 𝑜𝑘 = 𝑜)
67 eqidd 2799 . . . . . . . . . . . . 13 (𝑘 = 𝑜 → (𝑚𝑔𝑛) = (𝑚𝑔𝑛))
6866, 67goaleq12d 32747 . . . . . . . . . . . 12 (𝑘 = 𝑜 → ∀𝑔𝑘(𝑚𝑔𝑛) = ∀𝑔𝑜(𝑚𝑔𝑛))
6968eqeq2d 2809 . . . . . . . . . . 11 (𝑘 = 𝑜 → (𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛) ↔ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)))
7065, 69orbi12d 916 . . . . . . . . . 10 (𝑘 = 𝑜 → ((∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))))
7170cbvrexvw 3397 . . . . . . . . 9 (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)) ↔ ∃𝑜 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)))
723ne0ii 4253 . . . . . . . . . . . 12 ω ≠ ∅
73 r19.44zv 4407 . . . . . . . . . . . 12 (ω ≠ ∅ → (∃𝑙 ∈ ω (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))))
7472, 73ax-mp 5 . . . . . . . . . . 11 (∃𝑙 ∈ ω (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) ↔ (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)))
75 eqeq1 2802 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑚𝑔𝑛) → (𝑦 = (𝑖𝑔𝑗) ↔ (𝑚𝑔𝑛) = (𝑖𝑔𝑗)))
76752rexbidv 3259 . . . . . . . . . . . . . . 15 (𝑦 = (𝑚𝑔𝑛) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑖𝑔𝑗)))
77 ovexd 7175 . . . . . . . . . . . . . . 15 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) → (𝑚𝑔𝑛) ∈ V)
78 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → 𝑚 ∈ ω)
7943eqeq2d 2809 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑚 → ((𝑚𝑔𝑛) = (𝑖𝑔𝑗) ↔ (𝑚𝑔𝑛) = (𝑚𝑔𝑗)))
8079rexbidv 3256 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑚 → (∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑖𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑚𝑔𝑗)))
8180adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑖 = 𝑚) → (∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑖𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑚𝑔𝑗)))
82 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → 𝑛 ∈ ω)
8352eqeq2d 2809 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑛 → ((𝑚𝑔𝑛) = (𝑚𝑔𝑗) ↔ (𝑚𝑔𝑛) = (𝑚𝑔𝑛)))
8483adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑗 = 𝑛) → ((𝑚𝑔𝑛) = (𝑚𝑔𝑗) ↔ (𝑚𝑔𝑛) = (𝑚𝑔𝑛)))
85 eqidd 2799 . . . . . . . . . . . . . . . . . 18 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑔𝑛) = (𝑚𝑔𝑛))
8682, 84, 85rspcedvd 3574 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑚𝑔𝑗))
8778, 81, 86rspcedvd 3574 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑖𝑔𝑗))
8887ad5ant12 755 . . . . . . . . . . . . . . 15 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑚𝑔𝑛) = (𝑖𝑔𝑗))
8976, 77, 88elrabd 3630 . . . . . . . . . . . . . 14 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) → (𝑚𝑔𝑛) ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)})
90 oveq1 7147 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑚𝑔𝑛) → (𝑝𝑔𝑞) = ((𝑚𝑔𝑛)⊼𝑔𝑞))
9190eqeq2d 2809 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑚𝑔𝑛) → (𝑥 = (𝑝𝑔𝑞) ↔ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞)))
9291rexbidv 3256 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑚𝑔𝑛) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ↔ ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞)))
93 eqidd 2799 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑚𝑔𝑛) → 𝑘 = 𝑘)
94 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑚𝑔𝑛) → 𝑝 = (𝑚𝑔𝑛))
9593, 94goaleq12d 32747 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑚𝑔𝑛) → ∀𝑔𝑘𝑝 = ∀𝑔𝑘(𝑚𝑔𝑛))
9695eqeq2d 2809 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑚𝑔𝑛) → (𝑥 = ∀𝑔𝑘𝑝𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
9796rexbidv 3256 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑚𝑔𝑛) → (∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝 ↔ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
9892, 97orbi12d 916 . . . . . . . . . . . . . . 15 (𝑝 = (𝑚𝑔𝑛) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))))
9998adantl 485 . . . . . . . . . . . . . 14 ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) ∧ 𝑝 = (𝑚𝑔𝑛)) → ((∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))))
100 ovexd 7175 . . . . . . . . . . . . . . . . . . . 20 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜𝑔𝑙) ∈ V)
101 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → 𝑜 ∈ ω)
102101adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → 𝑜 ∈ ω)
103 oveq1 7147 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑜 → (𝑖𝑔𝑗) = (𝑜𝑔𝑗))
104103eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑜 → ((𝑜𝑔𝑙) = (𝑖𝑔𝑗) ↔ (𝑜𝑔𝑙) = (𝑜𝑔𝑗)))
105104rexbidv 3256 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑜 → (∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑖𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑜𝑔𝑗)))
106105adantl 485 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑖 = 𝑜) → (∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑖𝑔𝑗) ↔ ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑜𝑔𝑗)))
107 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → 𝑙 ∈ ω)
108 oveq2 7148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = 𝑙 → (𝑜𝑔𝑗) = (𝑜𝑔𝑙))
109108eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = 𝑙 → ((𝑜𝑔𝑙) = (𝑜𝑔𝑗) ↔ (𝑜𝑔𝑙) = (𝑜𝑔𝑙)))
110109adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑗 = 𝑙) → ((𝑜𝑔𝑙) = (𝑜𝑔𝑗) ↔ (𝑜𝑔𝑙) = (𝑜𝑔𝑙)))
111 eqidd 2799 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜𝑔𝑙) = (𝑜𝑔𝑙))
112107, 110, 111rspcedvd 3574 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑜𝑔𝑗))
113102, 106, 112rspcedvd 3574 . . . . . . . . . . . . . . . . . . . 20 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑖𝑔𝑗))
114 eqeq1 2802 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑜𝑔𝑙) → (𝑝 = (𝑖𝑔𝑗) ↔ (𝑜𝑔𝑙) = (𝑖𝑔𝑗)))
1151142rexbidv 3259 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑜𝑔𝑙) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑖𝑔𝑗)))
116 fmla0 32778 . . . . . . . . . . . . . . . . . . . . 21 (Fmla‘∅) = {𝑝 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑝 = (𝑖𝑔𝑗)}
117115, 116elrab2 3631 . . . . . . . . . . . . . . . . . . . 20 ((𝑜𝑔𝑙) ∈ (Fmla‘∅) ↔ ((𝑜𝑔𝑙) ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑜𝑔𝑙) = (𝑖𝑔𝑗)))
118100, 113, 117sylanbrc 586 . . . . . . . . . . . . . . . . . . 19 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑜𝑔𝑙) ∈ (Fmla‘∅))
119118adantr 484 . . . . . . . . . . . . . . . . . 18 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))) → (𝑜𝑔𝑙) ∈ (Fmla‘∅))
120 oveq2 7148 . . . . . . . . . . . . . . . . . . . 20 (𝑞 = (𝑜𝑔𝑙) → ((𝑚𝑔𝑛)⊼𝑔𝑞) = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)))
121120eqeq2d 2809 . . . . . . . . . . . . . . . . . . 19 (𝑞 = (𝑜𝑔𝑙) → (𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ↔ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))))
122121adantl 485 . . . . . . . . . . . . . . . . . 18 ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))) ∧ 𝑞 = (𝑜𝑔𝑙)) → (𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ↔ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))))
123 simpr 488 . . . . . . . . . . . . . . . . . 18 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))) → 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)))
124119, 122, 123rspcedvd 3574 . . . . . . . . . . . . . . . . 17 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙))) → ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞))
125124ex 416 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) → ∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞)))
126102adantr 484 . . . . . . . . . . . . . . . . . 18 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → 𝑜 ∈ ω)
12769adantl 485 . . . . . . . . . . . . . . . . . 18 ((((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) ∧ 𝑘 = 𝑜) → (𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛) ↔ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)))
128 simpr 488 . . . . . . . . . . . . . . . . . 18 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))
129126, 127, 128rspcedvd 3574 . . . . . . . . . . . . . . . . 17 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))
130129ex 416 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → (𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛) → ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
131125, 130orim12d 962 . . . . . . . . . . . . . . 15 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ((𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛))))
132131imp 410 . . . . . . . . . . . . . 14 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) → (∃𝑞 ∈ (Fmla‘∅)𝑥 = ((𝑚𝑔𝑛)⊼𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)))
13389, 99, 132rspcedvd 3574 . . . . . . . . . . . . 13 (((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) ∧ (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛))) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
134133ex 416 . . . . . . . . . . . 12 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) ∧ 𝑙 ∈ ω) → ((𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
135134rexlimdva 3243 . . . . . . . . . . 11 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → (∃𝑙 ∈ ω (𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
13674, 135syl5bir 246 . . . . . . . . . 10 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑜 ∈ ω) → ((∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
137136rexlimdva 3243 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑜 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑜𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑜(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
13871, 137syl5bi 245 . . . . . . . 8 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)))
139138rexlimivv 3251 . . . . . . 7 (∃𝑚 ∈ ω ∃𝑛 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑚𝑔𝑛)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑚𝑔𝑛)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
14061, 139sylbi 220 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)) → ∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝))
14142, 140impbii 212 . . . . 5 (∃𝑝 ∈ {𝑦 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑦 = (𝑖𝑔𝑗)} (∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
1428, 141bitri 278 . . . 4 (∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗)))
143142abbii 2863 . . 3 {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)} = {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))}
1446, 143uneq12i 4088 . 2 ((Fmla‘∅) ∪ {𝑥 ∣ ∃𝑝 ∈ (Fmla‘∅)(∃𝑞 ∈ (Fmla‘∅)𝑥 = (𝑝𝑔𝑞) ∨ ∃𝑘 ∈ ω 𝑥 = ∀𝑔𝑘𝑝)}) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
1452, 5, 1443eqtri 2825 1 (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3441   ∪ cun 3879  ∅c0 4243  {csn 4525   × cxp 5518  suc csuc 6164  ‘cfv 6327  (class class class)co 7140  ωcom 7567  1oc1o 8085  ∈𝑔cgoe 32729  ⊼𝑔cgna 32730  ∀𝑔cgol 32731  Fmlacfmla 32733 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-inf2 9095 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-map 8398  df-goel 32736  df-goal 32738  df-sat 32739  df-fmla 32741 This theorem is referenced by:  2goelgoanfmla1  32820
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