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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 6142 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
2 | csbvarg 3955 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
3 | 2 | fveq2d 6107 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
4 | 1, 3 | eqtrd 2644 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | csbprc 3932 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
6 | fvprc 6097 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
8 | 4, 7 | pm2.61i 175 | 1 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 ∅c0 3874 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 ax-pow 4769 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 |
This theorem is referenced by: mptcoe1fsupp 19406 mptcoe1matfsupp 20426 mp2pm2mplem4 20433 chfacfscmulfsupp 20483 chfacfpmmulfsupp 20487 cpmidpmatlem3 20496 cayhamlem4 20512 cayleyhamilton1 20516 logbmpt 24326 iuninc 28761 disjxpin 28783 finixpnum 32564 cdlemkid3N 35239 cdlemkid4 35240 cdlemk39s 35245 mccllem 38664 nbgrcl 40559 nbgrnvtx0 40563 |
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