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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 6956 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘⦋𝐴 / 𝑥⦌𝑥)) | |
2 | csbvarg 4440 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) | |
3 | 2 | fveq2d 6911 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘⦋𝐴 / 𝑥⦌𝑥) = (𝐹‘𝐴)) |
4 | 1, 3 | eqtrd 2775 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
5 | csbprc 4415 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = ∅) | |
6 | fvprc 6899 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) | |
7 | 5, 6 | eqtr4d 2778 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ⦋csb 3908 ∅c0 4339 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 |
This theorem is referenced by: mptcoe1fsupp 22233 mptcoe1matfsupp 22824 mp2pm2mplem4 22831 chfacfscmulfsupp 22881 chfacfpmmulfsupp 22885 cpmidpmatlem3 22894 cayhamlem4 22910 cayleyhamilton1 22914 logbmpt 26846 nbgrcl 29367 nbgrnvtx0 29371 iuninc 32581 disjxpin 32608 finixpnum 37592 cdlemkid3N 40916 cdlemkid4 40917 cdlemk39s 40922 mccllem 45553 clnbgrcl 47746 clnbgrnvtx0 47752 |
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