Theorem fvopabgf 3778

Description: The value of a function given by ordered-pair class 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 groups:   z,A   y,B   z,C   x,y   x,z

Proof of Theorem fvopabgf

Step	Hyp	Ref	Expression
1		ax-17 969	. . . . 5 |- (z e. w -> A.x z e. w)
2		fvopabgf.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		hbs1 1330	. . . . . . 7 |- ([w / x]u e. B -> A.x[w / x]u e. B)
6	5	hbab 1465	. . . . . 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 1562	. . . . 5 |- ({u | [w / x]u e. B} = C -> A.x{u | [w / x]u e. B} = C)
9		sbab 1580	. . . . . 6 |- (x = w -> B = {u | [w / x]u e. B})
10		fvopabgf.3	. . . . . 6 |- (x = A -> B = C)
11	9, 10	sylan9req 1525	. . . . 5 |- ((x = w /\ x = A) -> {u | [w / x]u e. B} = C)
12	8, 11	19.23ai 1062	. . . 4 |- (E.x(x = w /\ x = A) -> {u | [w / x]u e. B} = C)
13	4, 12	sylbi 199	. . 3 |- (w = A -> {u | [w / x]u e. B} = C)
14	13	fvopabg 3776	. 2 |- ((A e. D /\ C e. R) -> ({<.w, v>. | v = {u | [w / x]u e. B}}` A) = C)
15		ax-17 969	. . . 4 |- (y = B -> A.w y = B)
16		ax-17 969	. . . 4 |- (y = B -> A.v y = B)
17	6	hbeleq 1564	. . . 4 |- (v = {u | [w / x]u e. B} -> A.x v = {u | [w / x]u e. B})
18		ax-17 969	. . . 4 |- (v = {u | [w / x]u e. B} -> A.y v = {u | [w / x]u e. B})
19		id 59	. . . . 5 |- (y = v -> y = v)
20	19, 9	eqeqan12rd 1488	. . . 4 |- ((x = w /\ y = v) -> (y = B <-> v = {u | [w / x]u e. B}))
21	15, 16, 17, 18, 20	cbvopab 2667	. . 3 |- {<.x, y>. | y = B} = {<.w, v>. | v = {u | [w / x]u e. B}}
22	21	fveq1i 3716	. 2 |- ({<.x, y>. | y = B}` A) = ({<.w, v>. | v = {u | [w / x]u e. B}}` A)
23	14, 22	syl5eq 1516	1 |- ((A e. D /\ C e. R) -> ({<.x, y>. | y = B}` A) = C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168  {cab 1461  {copab 2661  ` cfv 3177

This theorem is referenced by:  fvopabf 3780  rdgsucopab 3937  frsucopab 3945

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-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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