Theorem csbfv12 6924

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 6524	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)
2		sbcbr123 5173	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)
3		csbconstg 3893	. . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	3	breq2d 5131	. . . . . 6 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
5	2, 4	bitrid 283	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	iotabidv 6515	. . . 4 ⊢ (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
7	1, 6	eqtrid 2782	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
8		df-fv 6539	. . . 4 ⊢ (𝐹‘𝐵) = (℩𝑦𝐵𝐹𝑦)
9	8	csbeq2i 3882	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦)
10		df-fv 6539	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)
11	7, 9, 10	3eqtr4g 2795	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
12		csbprc 4384	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∅)
13		csbprc 4384	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
14	13	fveq1d 6878	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (∅‘⦋𝐴 / 𝑥⦌𝐵))
15		0fv 6920	. . . 4 ⊢ (∅‘⦋𝐴 / 𝑥⦌𝐵) = ∅
16	14, 15	eqtr2di 2787	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
17	12, 16	eqtrd 2770	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
18	11, 17	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3459  [wsbc 3765  ⦋csb 3874  ∅c0 4308   class class class wbr 5119  ℩cio 6482  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-dm 5664  df-iota 6484  df-fv 6539

This theorem is referenced by:  csbfv2g  6925  coe1fzgsumdlem  22241  evl1gsumdlem  22294  csbrdgg  37347  rdgeqoa  37388  csbfinxpg  37406  cdlemk42  40960  evl1gprodd  42130  iccelpart  47447