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Mirrors > Home > ILE Home > Th. List > dmmpti | GIF version |
Description: Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fnmpti.1 | ⊢ 𝐵 ∈ V |
fnmpti.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
dmmpti | ⊢ dom 𝐹 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpti.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | fnmpti.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fnmpti 5301 | . 2 ⊢ 𝐹 Fn 𝐴 |
4 | fndm 5272 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ dom 𝐹 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 Vcvv 2712 ↦ cmpt 4028 dom cdm 4589 Fn wfn 5168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-fun 5175 df-fn 5176 |
This theorem is referenced by: brtpos2 6201 dmtopon 12517 |
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