Theorem tz6.12f 5719

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 3900	. . . . 5 |- ( z = y -> <. A , z >. = <. A , y >. )
2	1	eleq1d 2307	. . . 4 |- ( z = y -> ( <. A , z >. e. F <-> <. A , y >. e. F ) )
3		tz6.12f.1	. . . . . . 7 |- F/_ y F
4	3	nfel2 2405	. . . . . 6 |- F/ y <. A , z >. e. F
5		nfv 1581	. . . . . 6 |- F/ z <. A , y >. e. F
6	4, 5, 2	cbveu 2110	. . . . 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 477	. . 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 2248	. . 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 5718	. 2 |- ( ( <. A , z >. e. F /\ E! z <. A , z >. e. F ) -> ( F ` A ) = z )
12	10, 11	chvarv 1997	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E!weu 2086   e. wcel 2209   F/_wnfc 2379   <.cop 3708   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by: (None)