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Theorem fnex 5608
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5607. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnex ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Proof of Theorem fnex
StepHypRef Expression
1 fnrel 5189 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 272 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 df-fn 5094 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4 eleq1a 2187 . . . . . 6 (𝐴𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹𝐵))
54impcom 124 . . . . 5 ((dom 𝐹 = 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 resfunexg 5607 . . . . 5 ((Fun 𝐹 ∧ dom 𝐹𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
75, 6sylan2 282 . . . 4 ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴𝐴𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
87anassrs 395 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
93, 8sylanb 280 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10 resdm 4826 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2184 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
1211biimpa 292 . 2 ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
132, 9, 12syl2anc 406 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  Vcvv 2658  dom cdm 4507  cres 4509  Rel wrel 4512  Fun wfun 5085   Fn wfn 5086
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by:  funex  5609  fex  5613  offval  5955  ofrfval  5956  tfrlemibex  6192  tfr1onlembex  6208  fndmeng  6670  frecfzennn  10150
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