Theorem rdgeq2 3935

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 2364	. . . . . . . . . . 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 1486	. . . . . . . . . 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 2670	. . . . . . . . 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 3726	. . . . . . . 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 1486	. . . . . . 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 1663	. . . . . 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 616	. . . . 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 1664	. . . 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 1577	. . 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 2512	. 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 3932	. 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 3932	. 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 1531	1 |- (A = B -> rec(F, A) = rec(F, B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956  {cab 1463  A.wral 1645  E.wrex 1646  (/)c0 2280  ifcif 2361  U.cuni 2503  {copab 2666  Oncon0 2948  Lim wlim 2949  dom cdm 3170  ran crn 3171   |` cres 3172   Fn wfn 3177  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  rdg0t 3944  oav 4150  seq1val 6312

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-rdg 3932

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