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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 3819 | . 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
2 | fvmpts.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) | |
3 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3841 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3830 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 3, 4, 5 | cbvmpt 5161 | . . 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 6821 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⦋csb 3816 ↦ cmpt 5140 ‘cfv 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-iota 6343 df-fun 6387 df-fv 6393 |
This theorem is referenced by: fvmptdf 6829 fvmpocurryd 8018 mptnn0fsupp 13575 mptnn0fsuppr 13577 zsum 15287 prodss 15514 fprodser 15516 fprodn0 15546 fprodefsum 15661 pcmpt 16450 issubc 17346 gsummptnn0fz 19376 mptscmfsupp0 19969 gsummoncoe1 21230 fvmptnn04if 21751 prdsdsf 23270 itgparts 24949 dchrisumlema 26374 abfmpeld 30716 abfmpel 30717 cdlemk40 38673 aomclem6 40595 ellimcabssub0 42841 constlimc 42848 vonn0ioo2 43911 vonn0icc2 43913 |
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