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Mirrors > Home > MPE Home > Th. List > fvmpts | Structured version Visualization version GIF version |
Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fvmpts.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmpts | ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3896 | . 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
2 | fvmpts.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) | |
3 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3918 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3907 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 5259 | . . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2760 | . 2 ⊢ 𝐹 = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmptg 6996 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⦋csb 3893 ↦ cmpt 5231 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: fvmptdf 7004 fvmpocurryd 8255 mptnn0fsupp 13961 mptnn0fsuppr 13963 zsum 15663 prodss 15890 fprodser 15892 fprodn0 15922 fprodefsum 16037 pcmpt 16824 issubc 17784 gsummptnn0fz 19853 mptscmfsupp0 20536 gsummoncoe1 21827 fvmptnn04if 22350 prdsdsf 23872 itgparts 25563 dchrisumlema 26988 abfmpeld 31874 abfmpel 31875 cdlemk40 39783 aomclem6 41791 ellimcabssub0 44323 constlimc 44330 vonn0ioo2 45396 vonn0icc2 45398 |
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