Theorem rdgeq1 3992

Description: Equality theorem for the recursive definition generator.

Assertion

Ref	Expression
rdgeq1	⊢ (F = G → rec(F, A) = rec(G, A))

Proof of Theorem rdgeq1

Step	Hyp	Ref	Expression
1		fveq1 3780	. . . . . . . . . . . . 13 ⊢ (F = G → (F ‘(g ‘∪dom g)) = (G ‘(g ‘∪dom g)))
2	1	ifeq2d 2422	. . . . . . . . . . . 12 ⊢ (F = G → if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))) = if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))
3	2	ifeq2d 2422	. . . . . . . . . . 11 ⊢ (F = G → if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g)))) = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g)))))
4	3	eqeq2d 1533	. . . . . . . . . 10 ⊢ (F = G → (z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g)))) ↔ z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))))
5	4	opabbidv 2725	. . . . . . . . 9 ⊢ (F = G → {⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} = {⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))})
6	5	fveq1d 3783	. . . . . . . 8 ⊢ (F = G → ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))
7	6	eqeq2d 1533	. . . . . . 7 ⊢ (F = G → ((f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)) ↔ (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y))))
8	7	ralbidv 1710	. . . . . 6 ⊢ (F = G → (∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)) ↔ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y))))
9	8	anbi2d 627	. . . . 5 ⊢ (F = G → ((f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y))) ↔ (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))))
10	9	rexbidv 1711	. . . 4 ⊢ (F = G → (∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y))) ↔ ∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))))
11	10	abbidv 1624	. . 3 ⊢ (F = G → {f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)))} = {f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))})
12	11	unieqd 2566	. 2 ⊢ (F = G → ∪{f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)))} = ∪{f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))})
13		df-rdg 3990	. 2 ⊢ rec(F, A) = ∪{f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (F ‘(g ‘∪dom g))))} ‘(f ↾ y)))}
14		df-rdg 3990	. 2 ⊢ rec(G, A) = ∪{f∣∃x ∈ On (f Fn x ⋀ ∀y ∈ x (f ‘y) = ({⟨g, z⟩∣z = if(g = ∅, A, if(Lim dom g, ∪ran g, (G ‘(g ‘∪dom g))))} ‘(f ↾ y)))}
15	12, 13, 14	3eqtr4g 1578	1 ⊢ (F = G → rec(F, A) = rec(G, A))

Syntax hints:   → wi 3   ⋀ wa 230   = wceq 997  {cab 1509  ∀wral 1692  ∃wrex 1693  ∅c0 2331   ifcif 2413  ∪cuni 2557  {copab 2721  Oncon0 3005  Lim wlim 3006  dom cdm 3227  ran crn 3228   ↾ cres 3229   Fn wfn 3234   ‘cfv 3239  reccrdg 3989

This theorem is referenced by:  omv 4209  oev 4211  seq1val 6571  seq1suclem 6575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-cnv 3243  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fv 3255  df-rdg 3990

Copyright terms: Public domain