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Mirrors > Home > MPE Home > Th. List > csbfv | Structured version Visualization version GIF version |
Description: Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.) |
Ref | Expression |
---|---|
csbfv | ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbfv2g 6941 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
2 | csbvarg 4432 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
3 | 2 | fveq2d 6896 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
4 | 1, 3 | eqtrd 2773 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | csbprc 4407 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
6 | fvprc 6884 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | eqtr4d 2776 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
8 | 4, 7 | pm2.61i 182 | 1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⦋csb 3894 ∅c0 4323 ‘cfv 6544 |
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 5307 ax-pr 5428 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-dm 5687 df-iota 6496 df-fv 6552 |
This theorem is referenced by: mptcoe1fsupp 21739 mptcoe1matfsupp 22304 mp2pm2mplem4 22311 chfacfscmulfsupp 22361 chfacfpmmulfsupp 22365 cpmidpmatlem3 22374 cayhamlem4 22390 cayleyhamilton1 22394 logbmpt 26293 nbgrcl 28592 nbgrnvtx0 28596 iuninc 31792 disjxpin 31819 finixpnum 36473 cdlemkid3N 39804 cdlemkid4 39805 cdlemk39s 39810 mccllem 44313 |
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