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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 6945 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
2 | csbvarg 4433 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
3 | 2 | fveq2d 6900 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
4 | 1, 3 | eqtrd 2765 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | csbprc 4408 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
6 | fvprc 6888 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | eqtr4d 2768 | . 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 1533 ∈ wcel 2098 Vcvv 3461 ⦋csb 3889 ∅c0 4322 ‘cfv 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-dm 5688 df-iota 6501 df-fv 6557 |
This theorem is referenced by: mptcoe1fsupp 22163 mptcoe1matfsupp 22753 mp2pm2mplem4 22760 chfacfscmulfsupp 22810 chfacfpmmulfsupp 22814 cpmidpmatlem3 22823 cayhamlem4 22839 cayleyhamilton1 22843 logbmpt 26770 nbgrcl 29225 nbgrnvtx0 29229 iuninc 32435 disjxpin 32462 finixpnum 37211 cdlemkid3N 40538 cdlemkid4 40539 cdlemk39s 40544 mccllem 45125 clnbgrcl 47300 clnbgrnvtx0 47305 |
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