Theorem rdgeq2 4236

Description: Equality theorem for the recursive definition generator.

Assertion

Ref	Expression
rdgeq2	|- (A = B -> rec(F, A) = rec(F, B))

Proof of Theorem rdgeq2

Step	Hyp	Ref	Expression
1		ifeq1 2418	. . . . . . . . . . 11 |- (A = B -> if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))) = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g)))))
2	1	eqeq2d 1529	. . . . . . . . . 10 |- (A = B -> (z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g)))) <-> z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))))
3	2	opabbidv 2744	. . . . . . . . 9 |- (A = B -> {<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))} = {<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))})
4	3	fveq1d 3837	. . . . . . . 8 |- (A = B -> ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))
5	4	eqeq2d 1529	. . . . . . 7 |- (A = B -> ((f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)) <-> (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y))))
6	5	ralbidv 1709	. . . . . 6 |- (A = B -> (A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)) <-> A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y))))
7	6	anbi2d 619	. . . . 5 |- (A = B -> ((f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y))) <-> (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))))
8	7	rexbidv 1710	. . . 4 |- (A = B -> (E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y))) <-> E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))))
9	8	abbidv 1620	. . 3 |- (A = B -> {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))} = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))})
10	9	unieqd 2578	. 2 |- (A = B -> U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))} = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))})
11		df-rdg 4233	. 2 |- rec(F, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))}
12		df-rdg 4233	. 2 |- rec(F, B) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), B, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))}
13	10, 11, 12	3eqtr4g 1574	1 |- (A = B -> rec(F, A) = rec(F, B))

Syntax hints:   -> wi 3   /\ wa 221   = wceq 992  {cab 1505  A.wral 1691  E.wrex 1692  (/)c0 2332  ifcif 2415  U.cuni 2569  {copab 2740  Oncon0 2975  Lim wlim 2976  dom cdm 3251  ran crn 3252   |` cres 3253   Fn wfn 3258  ` cfv 3263  reccrdg 4232

This theorem is referenced by:  rdg0g 4245  oav 4286  seq1val 6677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-cnv 3267  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fv 3279  df-rdg 4233

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