Theorem dmfex 7945

Description: If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
dmfex	⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)

Proof of Theorem dmfex

Step	Hyp	Ref	Expression
1		fdm 6756	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
2		dmexg 7941	. . . 4 ⊢ (𝐹 ∈ 𝐶 → dom 𝐹 ∈ V)
3		eleq1 2832	. . . 4 ⊢ (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
4	2, 3	imbitrid 244	. . 3 ⊢ (dom 𝐹 = 𝐴 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V))
5	1, 4	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∈ 𝐶 → 𝐴 ∈ V))
6	5	impcom 407	1 ⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  dom cdm 5700  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711  df-fn 6576  df-f 6577

This theorem is referenced by:  wemoiso  8014  mapfset  8908  mapfoss  8910  fopwdom  9146  fsuppssov1  9453  fowdom  9640  wdomfil  10130  fin23lem17  10407  fin23lem32  10413  fin23lem39  10419  enfin1ai  10453  fin1a2lem7  10475  symgbasmap  19418  lindfmm  21870  kelac1  43020