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Theorem fex2 5086
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 4479 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
213adant1 933 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
3 fssxp 5085 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
433ad2ant1 936 . 2 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ⊆ (𝐴 × 𝐵))
52, 4ssexd 3924 1 ((𝐹:𝐴𝐵𝐴𝑉𝐵𝑊) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 896  wcel 1409  Vcvv 2574  wss 2944   × cxp 4370  wf 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-fun 4931  df-fn 4932  df-f 4933
This theorem is referenced by:  f1oen2g  6265  f1dom2g  6266  dom3d  6284  climrecvg1n  10097
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