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Mirrors > Home > MPE Home > Th. List > fvmpt2f | Structured version Visualization version GIF version |
Description: Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017.) |
Ref | Expression |
---|---|
fvmpt2f.0 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
fvmpt2f | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3831 | . . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵) | |
2 | csbid 3841 | . . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵 | |
3 | 1, 2 | eqtrdi 2849 | . 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
4 | fvmpt2f.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
6 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
7 | nfcsb1v 3852 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
8 | csbeq1a 3842 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 4, 5, 6, 7, 8 | cbvmptf 5129 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
10 | 3, 9 | fvmptg 6743 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 ⦋csb 3828 ↦ cmpt 5110 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: offval2f 7401 fmptcof2 30420 funcnvmpt 30430 esumc 31420 |
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