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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 3902 | . 2 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
| 2 | fvmpts.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) | |
| 3 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
| 4 | nfcsb1v 3923 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
| 5 | csbeq1a 3913 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
| 6 | 3, 4, 5 | cbvmpt 5253 | . . 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 7014 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⦋csb 3899 ↦ cmpt 5225 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: fvmptdf 7022 fvmpocurryd 8296 mptnn0fsupp 14038 mptnn0fsuppr 14040 zsum 15754 prodss 15983 fprodser 15985 fprodn0 16015 fprodefsum 16131 pcmpt 16930 issubc 17880 gsummptnn0fz 20004 mptscmfsupp0 20925 gsummoncoe1 22312 fvmptnn04if 22855 prdsdsf 24377 itgparts 26088 dchrisumlema 27532 abfmpeld 32664 abfmpel 32665 cdlemk40 40919 deg1gprod 42141 aomclem6 43071 ellimcabssub0 45632 constlimc 45639 vonn0ioo2 46705 vonn0icc2 46707 |
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