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Theorem fnex 6386
 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 6384. See fnexALT 7025 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 5903 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 480 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 df-fn 5807 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4 eleq1a 2683 . . . . . 6 (𝐴𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹𝐵))
54impcom 445 . . . . 5 ((dom 𝐹 = 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 resfunexg 6384 . . . . 5 ((Fun 𝐹 ∧ dom 𝐹𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
75, 6sylan2 490 . . . 4 ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴𝐴𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
87anassrs 678 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
93, 8sylanb 488 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10 resdm 5361 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2672 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
1211biimpa 500 . 2 ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
132, 9, 12syl2anc 691 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  dom cdm 5038   ↾ cres 5040  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  funex  6387  fex  6394  offval  6802  ofrfval  6803  suppvalfn  7189  suppfnss  7207  fnsuppeq0  7210  wfrlem15  7316  fndmeng  7919  fdmfifsupp  8168  cfsmolem  8975  axcc2lem  9141  unirnfdomd  9268  prdsbas2  15952  prdsplusgval  15956  prdsmulrval  15958  prdsleval  15960  prdsdsval  15961  prdsvscaval  15962  brssc  16297  sscpwex  16298  ssclem  16302  isssc  16303  rescval2  16311  reschom  16313  rescabs  16316  isfuncd  16348  dprdw  18232  prdsmgp  18433  dsmmbas2  19900  dsmmelbas  19902  ptval  21183  elptr  21186  prdstopn  21241  qtoptop  21313  imastopn  21333  vdgrfval  26422  suppss3  28890  ofcfval  29487  dya2iocuni  29672  trpredex  30981  stoweidlem27  38920  stoweidlem59  38952  omeiunle  39407
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