Theorem rdgeq2 3941

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 2368	. . . . . . . . . . 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 1489	. . . . . . . . . 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 2675	. . . . . . . . 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 3732	. . . . . . . 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 1489	. . . . . . 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 1666	. . . . . 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 618	. . . . 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 1667	. . . 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 1580	. . 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 2516	. 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 3938	. 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 3938	. 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 1534	1 |- (A = B -> rec(F, A) = rec(F, B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958  {cab 1466  A.wral 1648  E.wrex 1649  (/)c0 2283  ifcif 2365  U.cuni 2507  {copab 2671  Oncon0 2954  Lim wlim 2955  dom cdm 3176  ran crn 3177   |` cres 3178   Fn wfn 3183  ` cfv 3188  reccrdg 3937

This theorem is referenced by:  rdg0t 3950  oav 4156  seq1val 6313

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-rdg 3938

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