Theorem tz6.12f 5490

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 3738	. . . . 5 |- ( z = y -> <. A , z >. = <. A , y >. )
2	1	eleq1d 2223	. . . 4 |- ( z = y -> ( <. A , z >. e. F <-> <. A , y >. e. F ) )
3		tz6.12f.1	. . . . . . 7 |- F/_ y F
4	3	nfel2 2309	. . . . . 6 |- F/ y <. A , z >. e. F
5		nfv 1505	. . . . . 6 |- F/ z <. A , y >. e. F
6	4, 5, 2	cbveu 2027	. . . . 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 465	. . 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 2164	. . 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 5489	. 2 |- ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z )
12	10, 11	chvarv 1914	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E!weu 2003   e. wcel 2125   F/_wnfc 2283   <.cop 3559   ` cfv 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-iota 5128  df-fv 5171

This theorem is referenced by: (None)