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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 6955 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
| 2 | csbvarg 4434 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
| 3 | 2 | fveq2d 6910 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
| 4 | 1, 3 | eqtrd 2777 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
| 5 | csbprc 4409 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
| 6 | fvprc 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
| 7 | 5, 6 | eqtr4d 2780 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 ∅c0 4333 ‘cfv 6561 |
| 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 2708 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: mptcoe1fsupp 22217 mptcoe1matfsupp 22808 mp2pm2mplem4 22815 chfacfscmulfsupp 22865 chfacfpmmulfsupp 22869 cpmidpmatlem3 22878 cayhamlem4 22894 cayleyhamilton1 22898 logbmpt 26831 nbgrcl 29352 nbgrnvtx0 29356 iuninc 32573 disjxpin 32601 finixpnum 37612 cdlemkid3N 40935 cdlemkid4 40936 cdlemk39s 40941 mccllem 45612 clnbgrcl 47808 clnbgrnvtx0 47814 |
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