Theorem tz6.12f 5402

Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Hypothesis

Ref	Expression
tz6.12f.1	|- F/_ y F

Assertion

Ref	Expression
tz6.12f	|- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Distinct variable group:   y, A

Allowed substitution hint:   F( y)

Proof of Theorem tz6.12f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3670	. . . . 5 |- ( z = y -> <. A , z >. = <. A , y >. )
2	1	eleq1d 2181	. . . 4 |- ( z = y -> ( <. A , z >. e. F <-> <. A , y >. e. F ) )
3		tz6.12f.1	. . . . . . 7 |- F/_ y F
4	3	nfel2 2266	. . . . . 6 |- F/ y <. A , z >. e. F
5		nfv 1489	. . . . . 6 |- F/ z <. A , y >. e. F
6	4, 5, 2	cbveu 1997	. . . . 5 |- ( E! z <. A , z >. e. F <-> E! y <. A , y >. e. F )
7	6	a1i 9	. . . 4 |- ( z = y -> ( E! z <. A , z >. e. F <-> E! y <. A , y >. e. F ) )
8	2, 7	anbi12d 462	. . 3 |- ( z = y -> ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) <-> ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) ) )
9		eqeq2 2122	. . 3 |- ( z = y -> ( ( F ` A ) = z <-> ( F ` A ) = y ) )
10	8, 9	imbi12d 233	. 2 |- ( z = y -> ( ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z ) <-> ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y ) ) )
11		tz6.12 5401	. 2 |- ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z )
12	10, 11	chvarv 1885	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   E!weu 1973   F/_wnfc 2240   <.cop 3494   ` cfv 5079

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-iota 5044  df-fv 5087

This theorem is referenced by: (None)