Theorem dfafv2 41212

Description: Alternative definition of ( F''' A ) using ( F ` A ) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)

Assertion

Ref	Expression
dfafv2	|- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )

Proof of Theorem dfafv2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-afv 41197	. 2 |- ( F''' A ) = if ( F defAt A , ( iota x A F x ) , _V )
2		df-fv 5896	. . . 4 |- ( F ` A ) = ( iota x A F x )
3	2	eqcomi 2631	. . 3 |- ( iota x A F x ) = ( F ` A )
4		ifeq1 4090	. . 3 |- ( ( iota x A F x ) = ( F ` A ) -> if ( F defAt A , ( iota x A F x ) , _V ) = if ( F defAt A , ( F ` A ) , _V ) )
5	3, 4	ax-mp 5	. 2 |- if ( F defAt A , ( iota x A F x ) , _V ) = if ( F defAt A , ( F ` A ) , _V )
6	1, 5	eqtri 2644	1 |- ( F''' A ) = if ( F defAt A , ( F ` A ) , _V )

Syntax hints:   = wceq 1483   _Vcvv 3200   ifcif 4086   class class class wbr 4653   iotacio 5849   ` 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-rab 2921  df-v 3202  df-un 3579  df-if 4087  df-fv 5896  df-afv 41197

This theorem is referenced by:  afveq12d  41213  nfafv  41216  afvfundmfveq  41218  afvnfundmuv  41219  afvpcfv0  41226