Theorem csbfv12 6688

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 6317	. . . 4 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)
2		sbcbr123 5084	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)
3		csbconstg 3847	. . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	3	breq2d 5042	. . . . . 6 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
5	2, 4	syl5bb 286	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	iotabidv 6308	. . . 4 ⊢ (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
7	1, 6	syl5eq 2845	. . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦))
8		df-fv 6332	. . . 4 ⊢ (𝐹‘𝐵) = (℩𝑦𝐵𝐹𝑦)
9	8	csbeq2i 3836	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦)
10		df-fv 6332	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)
11	7, 9, 10	3eqtr4g 2858	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
12		csbprc 4313	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∅)
13		csbprc 4313	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
14	13	fveq1d 6647	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (∅‘⦋𝐴 / 𝑥⦌𝐵))
15		0fv 6684	. . . 4 ⊢ (∅‘⦋𝐴 / 𝑥⦌𝐵) = ∅
16	14, 15	eqtr2di 2850	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
17	12, 16	eqtrd 2833	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵))
18	11, 17	pm2.61i 185	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ⦋csb 3828  ∅c0 4243   class class class wbr 5030  ℩cio 6281  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174  ax-pow 5231

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-dm 5529  df-iota 6283  df-fv 6332

This theorem is referenced by:  csbfv2g  6689  coe1fzgsumdlem  20930  evl1gsumdlem  20980  csbwrecsg  34744  csbrdgg  34746  rdgeqoa  34787  csbfinxpg  34805  cdlemk42  38237  iccelpart  43950