Theorem tz6.12f 5544

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 3779	. . . . 5 |- ( z = y -> <. A , z >. = <. A , y >. )
2	1	eleq1d 2246	. . . 4 |- ( z = y -> ( <. A , z >. e. F <-> <. A , y >. e. F ) )
3		tz6.12f.1	. . . . . . 7 |- F/_ y F
4	3	nfel2 2332	. . . . . 6 |- F/ y <. A , z >. e. F
5		nfv 1528	. . . . . 6 |- F/ z <. A , y >. e. F
6	4, 5, 2	cbveu 2050	. . . . 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 473	. . 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 2187	. . 3 |- ( z = y -> ( ( F ` A ) = z <-> ( F ` A ) = y ) )
10	8, 9	imbi12d 234	. 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 5543	. 2 |- ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z )
12	10, 11	chvarv 1937	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E!weu 2026   e. wcel 2148   F/_wnfc 2306   <.cop 3595   ` cfv 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224

This theorem is referenced by: (None)