Theorem fvopab4gf 4744

Description: Value of a function given by an ordered-pair class abstraction. This version of fvopab4g 4742 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		fvopab4gf.1	. . . 4 |- (z e. A -> A.x z e. A)
2		fvopab4gf.2	. . . 4 |- (z e. C -> A.x z e. C)
3		fvopab4gf.3	. . . 4 |- (x = A -> B = C)
4	1, 2, 3	csbhypf 2572	. . 3 |- (w = A -> [_w / x]_B = C)
5		eqid 1884	. . 3 |- {<.w, v>. | (w e. D /\ v = [_w / x]_B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
6	4, 5	fvopab4g 4742	. 2 |- ((A e. D /\ C e. R) -> ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A) = C)
7		fvopab4gf.4	. . . 4 |- F = {<.x, y>. | (x e. D /\ y = B)}
8	7	fveq1i 4682	. . 3 |- (F` A) = ({<.x, y>. | (x e. D /\ y = B)}` A)
9		ax-17 1317	. . . . 5 |- ((x e. D /\ y = B) -> A.w(x e. D /\ y = B))
10		ax-17 1317	. . . . 5 |- ((x e. D /\ y = B) -> A.v(x e. D /\ y = B))
11		ax-17 1317	. . . . . 6 |- (w e. D -> A.x w e. D)
12		visset 2295	. . . . . . . 8 |- w e. _V
13		ax-17 1317	. . . . . . . 8 |- (v e. w -> A.x v e. w)
14	12, 13	hbcsb1 2568	. . . . . . 7 |- (v e. [_w / x]_B -> A.x v e. [_w / x]_B)
15	14	hbeleq 1997	. . . . . 6 |- (v = [_w / x]_B -> A.x v = [_w / x]_B)
16	11, 15	hban 1356	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.x(w e. D /\ v = [_w / x]_B))
17		ax-17 1317	. . . . 5 |- ((w e. D /\ v = [_w / x]_B) -> A.y(w e. D /\ v = [_w / x]_B))
18		eleq1 1957	. . . . . . 7 |- (x = w -> (x e. D <-> w e. D))
19	18	adantr 425	. . . . . 6 |- ((x = w /\ y = v) -> (x e. D <-> w e. D))
20		id 73	. . . . . . 7 |- (y = v -> y = v)
21		csbeq1a 2546	. . . . . . 7 |- (x = w -> B = [_w / x]_B)
22	20, 21	eqeqan12rd 1903	. . . . . 6 |- ((x = w /\ y = v) -> (y = B <-> v = [_w / x]_B))
23	19, 22	anbi12d 690	. . . . 5 |- ((x = w /\ y = v) -> ((x e. D /\ y = B) <-> (w e. D /\ v = [_w / x]_B)))
24	9, 10, 16, 17, 23	cbvopab 3403	. . . 4 |- {<.x, y>. | (x e. D /\ y = B)} = {<.w, v>. | (w e. D /\ v = [_w / x]_B)}
25	24	fveq1i 4682	. . 3 |- ({<.x, y>. | (x e. D /\ y = B)}` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
26	8, 25	eqtri 1908	. 2 |- (F` A) = ({<.w, v>. | (w e. D /\ v = [_w / x]_B)}` A)
27	6, 26	syl5eq 1940	1 |- ((A e. D /\ C e. R) -> (F` A) = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [_csb 2540  {copab 3395  ` cfv 3998

This theorem is referenced by:  fvopab4sf 4745  fnopabco 15711  sdclem2 15810  cncfres 15895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014

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