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Mirrors > Home > MPE Home > Th. List > fvmpt2i | Structured version Visualization version GIF version |
Description: Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
mptrcl.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmpt2i | ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3669 | . . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵) | |
2 | csbid 3674 | . . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵 | |
3 | 1, 2 | syl6eq 2802 | . 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
4 | mptrcl.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | nfcv 2894 | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
6 | nfcsb1v 3682 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
7 | csbeq1a 3675 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
8 | 5, 6, 7 | cbvmpt 4893 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
9 | 4, 8 | eqtri 2774 | . 2 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
10 | 3, 9 | fvmpti 6435 | 1 ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ⦋csb 3666 ↦ cmpt 4873 I cid 5165 ‘cfv 6041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ral 3047 df-rex 3048 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fv 6049 |
This theorem is referenced by: fvmpt2 6445 sumfc 14631 fsumf1o 14645 sumss 14646 isumshft 14762 prodfc 14866 fprodf1o 14867 mbfsup 23622 itg2splitlem 23706 dgrle 24190 |
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