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Theorem fex2 5338
Description: A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
fex2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)

Proof of Theorem fex2
StepHypRef Expression
1 xpexg 4700 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
213adant1 1000 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
3 fssxp 5337 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
433ad2ant1 1003 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ⊆ (𝐴 × 𝐵))
52, 4ssexd 4104 1 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 963  wcel 2128  Vcvv 2712  wss 3102   × cxp 4584  wf 5166
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-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
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 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-fun 5172  df-fn 5173  df-f 5174
This theorem is referenced by:  elmapg  6606  f1oen2g  6700  f1dom2g  6701  dom3d  6719  mapxpen  6793  climrecvg1n  11245  cnpfval  12595  txcn  12675  blfvalps  12785
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