Theorem frecseq123 8225

Description: Equality theorem for the well-founded recursion generator. (Contributed by Scott Fenton, 23-Dec-2021.)

Assertion

Ref	Expression
frecseq123	⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺))

Proof of Theorem frecseq123

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1138	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → 𝐴 = 𝐵)
2	1	sseq2d 3955	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
3		equid 2014	. . . . . . . . . . 11 ⊢ 𝑦 = 𝑦
4		predeq123 6260	. . . . . . . . . . 11 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑦 = 𝑦) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
5	3, 4	mp3an3 1453	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
6	5	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
7	6	sseq1d 3954	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
8	7	ralbidv 3161	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
9	2, 8	anbi12d 633	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥)))
10		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
11	10	oveqd 7377	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
12	6	reseq2d 5938	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
13	12	oveq2d 7376	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
14	11, 13	eqtrd 2772	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
15	14	eqeq2d 2748	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
16	15	ralbidv 3161	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
17	9, 16	3anbi23d 1442	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
18	17	exbidv 1923	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
19	18	abbidv 2803	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
20	19	unieqd 4864	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
21		df-frecs 8224	. 2 ⊢ frecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐹(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
22		df-frecs 8224	. 2 ⊢ frecs(𝑆, 𝐵, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))}
23	20, 21, 22	3eqtr4g 2797	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → frecs(𝑅, 𝐴, 𝐹) = frecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781  {cab 2715  ∀wral 3052   ⊆ wss 3890  ∪ cuni 4851   ↾ cres 5626  Predcpred 6258   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  frecscfrecs 8223

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fv 6500  df-ov 7363  df-frecs 8224

This theorem is referenced by:  wrecseq123  8256  csbwrecsg  8261