Theorem elfv2ex 6229

Description: If a function value of a function value has a member, the first argument is a set. (Contributed by AV, 8-Apr-2021.)

Assertion

Ref	Expression
elfv2ex	|- ( A e. ( ( F ` B ) ` C ) -> B e. _V )

Proof of Theorem elfv2ex

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 |- ( B e. _V -> ( A e. ( ( F ` B ) ` C ) -> B e. _V ) )
2		fv2prc 6228	. . . 4 |- ( -. B e. _V -> ( ( F ` B ) ` C ) = (/) )
3	2	eleq2d 2687	. . 3 |- ( -. B e. _V -> ( A e. ( ( F ` B ) ` C ) <-> A e. (/) ) )
4		noel 3919	. . . 4 |- -. A e. (/)
5	4	pm2.21i 116	. . 3 |- ( A e. (/) -> B e. _V )
6	3, 5	syl6bi 243	. 2 |- ( -. B e. _V -> ( A e. ( ( F ` B ) ` C ) -> B e. _V ) )
7	1, 6	pm2.61i 176	1 |- ( A e. ( ( F ` B ) ` C ) -> B e. _V )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` 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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)