Theorem csbfv12g 5337

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

Assertion

Ref	Expression
csbfv12g	|- (A e. C -> [_A / x]_(F` B) = ([_A / x]_F` [_A / x]_B))

Proof of Theorem csbfv12g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbiotag 4372	. . 3 |- (A e. C -> [_A / x]_(iotayBFy) = (iotay [.A / x ].BFy))
2		sbcbrg 4686	. . . . 5 |- (A e. C -> ( [.A / x ].BFy <-> [_A / x]_B[_A / x]_F[_A / x]_y))
3		csbconstg 3151	. . . . . 6 |- (A e. C -> [_A / x]_y = y)
4	3	breq2d 4652	. . . . 5 |- (A e. C -> ([_A / x]_B[_A / x]_F[_A / x]_y <-> [_A / x]_B[_A / x]_Fy))
5	2, 4	bitrd 244	. . . 4 |- (A e. C -> ( [.A / x ].BFy <-> [_A / x]_B[_A / x]_Fy))
6	5	iotabidv 4361	. . 3 |- (A e. C -> (iotay [.A / x ].BFy) = (iotay[_A / x]_B[_A / x]_Fy))
7	1, 6	eqtrd 2385	. 2 |- (A e. C -> [_A / x]_(iotayBFy) = (iotay[_A / x]_B[_A / x]_Fy))
8		df-fv 4796	. . 3 |- (F` B) = (iotayBFy)
9	8	csbeq2i 3163	. 2 |- [_A / x]_(F` B) = [_A / x]_(iotayBFy)
10		df-fv 4796	. 2 |- ([_A / x]_F` [_A / x]_B) = (iotay[_A / x]_B[_A / x]_Fy)
11	7, 9, 10	3eqtr4g 2410	1 |- (A e. C -> [_A / x]_(F` B) = ([_A / x]_F` [_A / x]_B))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710   [.wsbc 3047  [_csb 3137  iotacio 4338   class class class wbr 4640  ` cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  csbfv2g  5338