Theorem fvopabgf 2874

Description: The value of a function given by ordered pair abstraction.

Hypotheses

Ref	Expression
fvopabgf.1	|- (z e. A -> A.x z e. A)
fvopabgf.2	|- (z e. C -> A.x z e. C)
fvopabgf.3	|- (x = A -> B = C)

Assertion

Ref	Expression
fvopabgf	|- ((A e. D /\ C e. R) -> ({<.x, y>. | y = B}` A) = C)

Distinct variable group(s):   z,A   y,B   z,C   x,y   x,z

Proof of Theorem fvopabgf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (z e. w -> A.x z e. w)
2		fvopabgf.1	. . . . 5 |- (z e. A -> A.x z e. A)
3		visset 1350	. . . . 5 |- w e. V
4	1, 2, 3	eqvincf 1408	. . . 4 |- (w = A <-> E.x(x = w /\ x = A))
5		hbs1 986	. . . . . . 7 |- ([w / x]u e. B -> A.x[w / x]u e. B)
6	5	hbab 1096	. . . . . 6 |- (v e. {u | [w / x]u e. B} -> A.x v e. {u | [w / x]u e. B})
7		fvopabgf.2	. . . . . 6 |- (z e. C -> A.x z e. C)
8	6, 7	hbeq 1171	. . . . 5 |- ({u | [w / x]u e. B} = C -> A.x{u | [w / x]u e. B} = C)
9		sbab 1188	. . . . . . 7 |- (x = w -> B = {u | [w / x]u e. B})
10	9	cleqcomd 1106	. . . . . 6 |- (x = w -> {u | [w / x]u e. B} = B)
11		fvopabgf.3	. . . . . 6 |- (x = A -> B = C)
12	10, 11	sylan9eq 1144	. . . . 5 |- ((x = w /\ x = A) -> {u | [w / x]u e. B} = C)
13	8, 12	19.23ai 746	. . . 4 |- (E.x(x = w /\ x = A) -> {u | [w / x]u e. B} = C)
14	4, 13	sylbi 174	. . 3 |- (w = A -> {u | [w / x]u e. B} = C)
15	14	fvopabg 2872	. 2 |- ((A e. D /\ C e. R) -> ({<.w, v>. | v = {u | [w / x]u e. B}}` A) = C)
16		ax-17 925	. . . 4 |- (y = B -> A.w y = B)
17		ax-17 925	. . . 4 |- (y = B -> A.v y = B)
18	6	hbeleq 1173	. . . 4 |- (v = {u | [w / x]u e. B} -> A.x v = {u | [w / x]u e. B})
19		ax-17 925	. . . 4 |- (v = {u | [w / x]u e. B} -> A.y v = {u | [w / x]u e. B})
20		id 9	. . . . 5 |- (y = v -> y = v)
21	20, 9	cleqan12rd 1117	. . . 4 |- ((x = w /\ y = v) -> (y = B <-> v = {u | [w / x]u e. B}))
22	16, 17, 18, 19, 21	cbvopab 2104	. . 3 |- {<.x, y>. | y = B} = {<.w, v>. | v = {u | [w / x]u e. B}}
23	22	fveq1i 2833	. 2 |- ({<.x, y>. | y = B}` A) = ({<.w, v>. | v = {u | [w / x]u e. B}}` A)
24	15, 23	syl5eq 1136	1 |- ((A e. D /\ C e. R) -> ({<.x, y>. | y = B}` A) = C)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092  {copab 2055  ` cfv 2422

This theorem is referenced by:  fvopabf 2876  rdgsucopab 2984  frsucopab 2992

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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