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Mirrors > Home > MPE Home > Th. List > csbfv12 | Structured version Visualization version GIF version |
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.) |
Ref | Expression |
---|---|
csbfv12 | ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiota 6528 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) | |
2 | sbcbr123 5198 | . . . . . 6 ⊢ ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦) | |
3 | csbconstg 3910 | . . . . . . 7 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) | |
4 | 3 | breq2d 5156 | . . . . . 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 6519 | . . . 4 ⊢ (𝐴 ∈ V → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
7 | 1, 6 | eqtrid 2785 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
8 | df-fv 6543 | . . . 4 ⊢ (𝐹‘𝐵) = (℩𝑦𝐵𝐹𝑦) | |
9 | 8 | csbeq2i 3899 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) |
10 | df-fv 6543 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦) | |
11 | 7, 9, 10 | 3eqtr4g 2798 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
12 | csbprc 4404 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ∅) | |
13 | csbprc 4404 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅) | |
14 | 13 | fveq1d 6883 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (∅‘⦋𝐴 / 𝑥⦌𝐵)) |
15 | 0fv 6925 | . . . 4 ⊢ (∅‘⦋𝐴 / 𝑥⦌𝐵) = ∅ | |
16 | 14, 15 | eqtr2di 2790 | . . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
17 | 12, 16 | eqtrd 2773 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 [wsbc 3775 ⦋csb 3891 ∅c0 4320 class class class wbr 5144 ℩cio 6485 ‘cfv 6535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-dm 5682 df-iota 6487 df-fv 6543 |
This theorem is referenced by: csbfv2g 6930 coe1fzgsumdlem 21794 evl1gsumdlem 21844 csbrdgg 36115 rdgeqoa 36156 csbfinxpg 36174 cdlemk42 39718 iccelpart 45974 |
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