Theorem fvopabnf 3783

Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvopabn 3781 uses bound-variable hypotheses instead of distinct variable conditions.

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
fvopabnf	|- (-. C e. V -> ({<.x, y>. | y = B}` A) = (/))

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

Proof of Theorem fvopabnf

Step	Hyp	Ref	Expression
1		ax-17 970	. . . . 5 |- (z e. w -> A.x z e. w)
2		fvopabgf.1	. . . . 5 |- (z e. A -> A.x z e. A)
3		visset 1810	. . . . 5 |- w e. V
4	1, 2, 3	eqvincf 1881	. . . 4 |- (w = A <-> E.x(x = w /\ x = A))
5		hbs1 1331	. . . . . . 7 |- ([w / x]u e. B -> A.x[w / x]u e. B)
6	5	hbab 1466	. . . . . 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 1563	. . . . 5 |- ({u | [w / x]u e. B} = C -> A.x{u | [w / x]u e. B} = C)
9		sbab 1581	. . . . . 6 |- (x = w -> B = {u | [w / x]u e. B})
10		fvopabgf.3	. . . . . 6 |- (x = A -> B = C)
11	9, 10	sylan9req 1526	. . . . 5 |- ((x = w /\ x = A) -> {u | [w / x]u e. B} = C)
12	8, 11	19.23ai 1063	. . . 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	fvopabn 3781	. 2 |- (-. C e. V -> ({<.w, v>. | v = {u | [w / x]u e. B}}` A) = (/))
15		ax-17 970	. . . 4 |- (y = B -> A.w y = B)
16		ax-17 970	. . . 4 |- (y = B -> A.v y = B)
17	6	hbeleq 1565	. . . 4 |- (v = {u | [w / x]u e. B} -> A.x v = {u | [w / x]u e. B})
18		ax-17 970	. . . 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 1489	. . . 4 |- ((x = w /\ y = v) -> (y = B <-> v = {u | [w / x]u e. B}))
21	15, 16, 17, 18, 20	cbvopab 2668	. . 3 |- {<.x, y>. | y = B} = {<.w, v>. | v = {u | [w / x]u e. B}}
22	21	fveq1i 3720	. 2 |- ({<.x, y>. | y = B}` A) = ({<.w, v>. | v = {u | [w / x]u e. B}}` A)
23	14, 22	syl5eq 1517	1 |- (-. C e. V -> ({<.x, y>. | y = B}` A) = (/))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  [wsbc 1169  {cab 1462  Vcvv 1808  (/)c0 2277  {copab 2662  ` cfv 3178

This theorem is referenced by:  rdgsucopabn 3942

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194

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