Theorem dmmpti 5508

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.)

Hypotheses

Ref	Expression
fnmpti.1	⊢ 𝐵 ∈ V
fnmpti.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
dmmpti	⊢ dom 𝐹 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem dmmpti

Step	Hyp	Ref	Expression
1		fnmpti.1	. . 3 ⊢ 𝐵 ∈ V
2		fnmpti.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 5507	. 2 ⊢ 𝐹 Fn 𝐴
4		fndm 5475	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	ax-mp 5	1 ⊢ dom 𝐹 = 𝐴

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4187  dom cdm 4769   Fn wfn 5367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-fun 5374  df-fn 5375

This theorem is referenced by:  brtpos2  6512  ballotfilem7  13257  dmtopon  15047