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Theorem fnex 6466
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 6464. See fnexALT 7117 for alternate proof. (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 5977 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 481 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 df-fn 5879 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4 eleq1a 2694 . . . . . 6 (𝐴𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹𝐵))
54impcom 446 . . . . 5 ((dom 𝐹 = 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 resfunexg 6464 . . . . 5 ((Fun 𝐹 ∧ dom 𝐹𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
75, 6sylan2 491 . . . 4 ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴𝐴𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
87anassrs 679 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
93, 8sylanb 489 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10 resdm 5429 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2684 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
1211biimpa 501 . 2 ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
132, 9, 12syl2anc 692 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  dom cdm 5104  cres 5106  Rel wrel 5109  Fun wfun 5870   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by:  funex  6467  fex  6475  offval  6889  ofrfval  6890  suppvalfn  7287  suppfnss  7305  fnsuppeq0  7308  wfrlem15  7414  fndmeng  8019  fdmfifsupp  8270  cfsmolem  9077  axcc2lem  9243  unirnfdomd  9374  prdsbas2  16110  prdsplusgval  16114  prdsmulrval  16116  prdsleval  16118  prdsdsval  16119  prdsvscaval  16120  brssc  16455  sscpwex  16456  ssclem  16460  isssc  16461  rescval2  16469  reschom  16471  rescabs  16474  isfuncd  16506  dprdw  18390  prdsmgp  18591  dsmmbas2  20062  dsmmelbas  20064  ptval  21354  elptr  21357  prdstopn  21412  qtoptop  21484  imastopn  21504  suppss3  29476  ofcfval  30134  dya2iocuni  30319  trpredex  31711  fnexd  39143  stoweidlem27  40007  stoweidlem59  40039  omeiunle  40494
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