Theorem csbfv12 6231

Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)

Assertion

Ref	Expression
csbfv12	|- [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B )

Proof of Theorem csbfv12

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbiota 5881	. . . 4 |- [_ A / x ]_ ( iota y B F y ) = ( iota y [. A / x ]. B F y )
2		sbcbr123 4706	. . . . . 6 |- ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y )
3		csbconstg 3546	. . . . . . 7 |- ( A e. _V -> [_ A / x ]_ y = y )
4	3	breq2d 4665	. . . . . 6 |- ( A e. _V -> ( [_ A / x ]_ B [_ A / x ]_ F [_ A / x ]_ y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
5	2, 4	syl5bb 272	. . . . 5 |- ( A e. _V -> ( [. A / x ]. B F y <-> [_ A / x ]_ B [_ A / x ]_ F y ) )
6	5	iotabidv 5872	. . . 4 |- ( A e. _V -> ( iota y [. A / x ]. B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
7	1, 6	syl5eq 2668	. . 3 |- ( A e. _V -> [_ A / x ]_ ( iota y B F y ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y ) )
8		df-fv 5896	. . . 4 |- ( F ` B ) = ( iota y B F y )
9	8	csbeq2i 3993	. . 3 |- [_ A / x ]_ ( F ` B ) = [_ A / x ]_ ( iota y B F y )
10		df-fv 5896	. . 3 |- ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( iota y [_ A / x ]_ B [_ A / x ]_ F y )
11	7, 9, 10	3eqtr4g 2681	. 2 |- ( A e. _V -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
12		csbprc 3980	. . 3 |- ( -. A e. _V -> [_ A / x ]_ ( F ` B ) = (/) )
13		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ F = (/) )
14	13	fveq1d 6193	. . . 4 |- ( -. A e. _V -> ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( (/) ` [_ A / x ]_ B ) )
15		0fv 6227	. . . 4 |- ( (/) ` [_ A / x ]_ B ) = (/)
16	14, 15	syl6req 2673	. . 3 |- ( -. A e. _V -> (/) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
17	12, 16	eqtrd 2656	. 2 |- ( -. A e. _V -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
18	11, 17	pm2.61i 176	1 |- [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   class class class wbr 4653   iotacio 5849   ` 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-fal 1489  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-sbc 3436  df-csb 3534  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:  csbfv2g  6232  coe1fzgsumdlem  19671  evl1gsumdlem  19720  csbwrecsg  33173  csbrdgg  33175  rdgeqoa  33218  csbfinxpg  33225  cdlemk42  36229  iccelpart  41369