Theorem tz6.12f 6212

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 4403	. . . . 5 |- ( z = y -> <. A , z >. = <. A , y >. )
2	1	eleq1d 2686	. . . 4 |- ( z = y -> ( <. A , z >. e. F <-> <. A , y >. e. F ) )
3		tz6.12f.1	. . . . . . 7 |- F/_ y F
4	3	nfel2 2781	. . . . . 6 |- F/ y <. A , z >. e. F
5		nfv 1843	. . . . . 6 |- F/ z <. A , y >. e. F
6	4, 5, 2	cbveu 2505	. . . . 5 |- ( E! z <. A , z >. e. F <-> E! y <. A , y >. e. F )
7	6	a1i 11	. . . 4 |- ( z = y -> ( E! z <. A , z >. e. F <-> E! y <. A , y >. e. F ) )
8	2, 7	anbi12d 747	. . 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 2633	. . 3 |- ( z = y -> ( ( F ` A ) = z <-> ( F ` A ) = y ) )
10	8, 9	imbi12d 334	. 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 6211	. 2 |- ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z )
12	10, 11	chvarv 2263	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   F/_wnfc 2751   <.cop 4183   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)