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Theorem dmfex 5401
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 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)

Proof of Theorem dmfex
StepHypRef Expression
1 fdm 5367 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
2 dmexg 4887 . . . 4 (𝐹𝐶 → dom 𝐹 ∈ V)
3 eleq1 2240 . . . 4 (dom 𝐹 = 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V))
42, 3imbitrid 154 . . 3 (dom 𝐹 = 𝐴 → (𝐹𝐶𝐴 ∈ V))
51, 4syl 14 . 2 (𝐹:𝐴𝐵 → (𝐹𝐶𝐴 ∈ V))
65impcom 125 1 ((𝐹𝐶𝐹:𝐴𝐵) → 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  dom cdm 4623  wf 5208
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-cnv 4631  df-dm 4633  df-rn 4634  df-fn 5215  df-f 5216
This theorem is referenced by:  fopwdom  6830
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