| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 3902 | . . 3 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵) | |
| 2 | csbid 3912 | . . 3 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵 | |
| 3 | 1, 2 | eqtrdi 2793 | . 2 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
| 4 | fvmpt2f.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 5 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 6 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 7 | nfcsb1v 3923 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
| 8 | csbeq1a 3913 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
| 9 | 4, 5, 6, 7, 8 | cbvmptf 5251 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
| 10 | 3, 9 | fvmptg 7014 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 ⦋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: offval2f 7712 fmptcof2 32667 funcnvmpt 32677 esumc 34052 fvmpt2df 45279 fvmpt4d 45283 smfpimltxrmptf 46773 smfpimgtxrmptf 46799 |
| Copyright terms: Public domain | W3C validator |