Theorem tz6.12 6211

Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Assertion

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

Distinct variable groups:   y, F   y, A

Proof of Theorem tz6.12

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A F y <-> <. A , y >. e. F )
2	1	eubii 2492	. 2 |- ( E! y A F y <-> E! y <. A , y >. e. F )
3		tz6.12-1 6210	. 2 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )
4	1, 2, 3	syl2anbr 497	1 |- ( ( <. A , y >. e. F /\ E! y <. A , y >. e. F ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   <.cop 4183   class class class wbr 4653   ` 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-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-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  tz6.12f  6212  dfac5lem5  8950  tz6.12-afv  41253