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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 3863 | . 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
2 | fvmpts.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) | |
3 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3885 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3874 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 5221 | . . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 2, 6 | eqtri 2765 | . 2 ⊢ 𝐹 = (𝑦 ∈ 𝐶 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 1, 7 | fvmptg 6951 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⦋csb 3860 ↦ cmpt 5193 ‘cfv 6501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 |
This theorem is referenced by: fvmptdf 6959 fvmpocurryd 8207 mptnn0fsupp 13909 mptnn0fsuppr 13911 zsum 15610 prodss 15837 fprodser 15839 fprodn0 15869 fprodefsum 15984 pcmpt 16771 issubc 17728 gsummptnn0fz 19770 mptscmfsupp0 20403 gsummoncoe1 21691 fvmptnn04if 22214 prdsdsf 23736 itgparts 25427 dchrisumlema 26852 abfmpeld 31612 abfmpel 31613 cdlemk40 39409 aomclem6 41415 ellimcabssub0 43932 constlimc 43939 vonn0ioo2 45005 vonn0icc2 45007 |
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