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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 3888 | . . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵) | |
2 | csbid 3898 | . . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵 | |
3 | 1, 2 | syl6eq 2874 | . 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
4 | fvmpt2f.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | nfcv 2979 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
6 | nfcv 2979 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
7 | nfcsb1v 3909 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
8 | csbeq1a 3899 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
9 | 4, 5, 6, 7, 8 | cbvmptf 5167 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
10 | 3, 9 | fvmptg 6768 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 ⦋csb 3885 ↦ cmpt 5148 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 |
This theorem is referenced by: offval2f 7423 fmptcof2 30404 funcnvmpt 30414 esumc 31312 |
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