Theorem afvnufveq 41227

Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvnufveq	|- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )

Proof of Theorem afvnufveq

Step	Hyp	Ref	Expression
1		afvfundmfveq 41218	. . . 4 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
2	1	con3i 150	. . 3 |- ( -. ( F''' A ) = ( F ` A ) -> -. F defAt A )
3		afvnfundmuv 41219	. . 3 |- ( -. F defAt A -> ( F''' A ) = _V )
4	2, 3	syl 17	. 2 |- ( -. ( F''' A ) = ( F ` A ) -> ( F''' A ) = _V )
5	4	necon1ai 2821	1 |- ( ( F''' A ) =/= _V -> ( F''' A ) = ( F ` A ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   =/= wne 2794   _Vcvv 3200   ` cfv 5888   defAt wdfat 41193  '''cafv 41194

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-afv 41197

This theorem is referenced by:  afvvfveq  41228  afvfv0bi  41232  aovnuoveq  41271