Theorem fvopab4gf 3772

Description: Value of a function given by an ordered-pair class abstraction. This version of fvopab4g 3770 uses bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
fvopab4gf.1	|- (z e. A -> A.x z e. A)
fvopab4gf.2	|- (z e. C -> A.x z e. C)
fvopab4gf.3	|- (x = A -> B = C)
fvopab4gf.4	|- F = {<.x, y>. | (x e. D /\ y = B)}

Assertion

Ref	Expression
fvopab4gf	|- ((A e. D /\ C e. R) -> (F` A) = C)

Distinct variable groups:   z,A   y,B   z,C   x,y,D   x,z

Proof of Theorem fvopab4gf

Step	Hyp	Ref	Expression
1		ax-17 969	. . . . 5 |- (z e. w -> A.x z e. w)
2		fvopab4gf.1	. . . . 5 |- (z e. A -> A.x z e. A)
3		visset 1809	. . . . 5 |- w e. V
4	1, 2, 3	eqvincf 1880	. . . 4 |- (w = A <-> E.x(x = w /\ x = A))
5		ax-17 969	. . . . . . 7 |- (v e. w -> A.x v e. w)
6	3, 5	hbcsb1 2021	. . . . . 6 |- (v e. [_w / x]_B -> A.x v e. [_w / x]_B)
7		fvopab4gf.2	. . . . . 6 |- (z e. C -> A.x z e. C)
8	6, 7	hbeq 1562	. . . . 5 |- ([_w / x]_B = C -> A.x[_w / x]_B = C)
9		csbeq1a 2002	. . . . . 6 |- (x = w -> B = [_w / x]_B)
10		fvopab4gf.3	. . . . . 6 |- (x = A -> B = C)
11	9, 10	sylan9req 1525	. . . . 5 |- ((x = w /\ x = A) -> [_w / x]_B = C)
12	8, 11	19.23ai 1062	. . . 4 |- (E.x(x = w /\ x = A) -> [_w / x]_B = C)
13	4, 12	sylbi 199	. . 3 |- (w = A -> [_w / x]_B = C)
14		eqid 1473	. . 3 |- {<.w, v>. | (w e. D /\ v = [_w / x]_B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
15	13, 14	fvopab4g 3770	. 2 |- ((A e. D /\ C e. R) -> ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A) = C)
16		fvopab4gf.4	. . . 4 |- F = {<.x, y>. | (x e. D /\ y = B)}
17	16	fveq1i 3716	. . 3 |- (F` A) = ({<.x, y>. | (x e. D /\ y = B)}` A)
18		ax-17 969	. . . . 5 |- ((x e. D /\ y = B) -> A.w(x e. D /\ y = B))
19		ax-17 969	. . . . 5 |- ((x e. D /\ y = B) -> A.v(x e. D /\ y = B))
20		ax-17 969	. . . . . 6 |- (w e. D -> A.x w e. D)
21	6	hbeleq 1564	. . . . . 6 |- (v = [_w / x]_B -> A.x v = [_w / x]_B)
22	20, 21	hban 1007	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.x(w e. D /\ v = [_w / x]_B))
23		ax-17 969	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.y(w e. D /\ v = [_w / x]_B))
24		eleq1 1531	. . . . . . 7 |- (x = w -> (x e. D <-> w e. D))
25	24	adantr 389	. . . . . 6 |- ((x = w /\ y = v) -> (x e. D <-> w e. D))
26		id 59	. . . . . . 7 |- (y = v -> y = v)
27	26, 9	eqeqan12rd 1488	. . . . . 6 |- ((x = w /\ y = v) -> (y = B <-> v = [_w / x]_B))
28	25, 27	anbi12d 627	. . . . 5 |- ((x = w /\ y = v) -> ((x e. D /\ y = B) <-> (w e. D /\ v = [_w / x]_B)))
29	18, 19, 22, 23, 28	cbvopab 2667	. . . 4 |- {<.x, y>. | (x e. D /\ y = B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
30	29	fveq1i 3716	. . 3 |- ({<.x, y>. | (x e. D /\ y = B)}` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
31	17, 30	eqtr 1492	. 2 |- (F` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
32	15, 31	syl5eq 1516	1 |- ((A e. D /\ C e. R) -> (F` A) = C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  [_csb 1997  {copab 2661  ` cfv 3177

This theorem is referenced by:  fvopab4sf 3773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193

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