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Theorem fnex 6268
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 6266. See fnexALT 6905 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 5793 . . 3 (𝐹 Fn 𝐴 → Rel 𝐹)
21adantr 479 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → Rel 𝐹)
3 df-fn 5697 . . 3 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4 eleq1a 2587 . . . . . 6 (𝐴𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹𝐵))
54impcom 444 . . . . 5 ((dom 𝐹 = 𝐴𝐴𝐵) → dom 𝐹𝐵)
6 resfunexg 6266 . . . . 5 ((Fun 𝐹 ∧ dom 𝐹𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
75, 6sylan2 489 . . . 4 ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴𝐴𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
87anassrs 677 . . 3 (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
93, 8sylanb 487 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10 resdm 5252 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
1110eleq1d 2576 . . 3 (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
1211biimpa 499 . 2 ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
132, 9, 12syl2anc 690 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  dom cdm 4932  cres 4934  Rel wrel 4937  Fun wfun 5688   Fn wfn 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702
This theorem is referenced by:  funex  6269  fex  6276  offval  6683  ofrfval  6684  suppvalfn  7069  suppfnss  7087  fnsuppeq0  7090  wfrlem15  7196  fndmeng  7799  fdmfifsupp  8048  cfsmolem  8855  axcc2lem  9021  unirnfdomd  9148  prdsbas2  15840  prdsplusgval  15844  prdsmulrval  15846  prdsleval  15848  prdsdsval  15849  prdsvscaval  15850  brssc  16193  sscpwex  16194  ssclem  16198  isssc  16199  rescval2  16207  reschom  16209  rescabs  16212  isfuncd  16244  dprdw  18143  prdsmgp  18344  dsmmbas2  19807  dsmmelbas  19809  ptval  21090  elptr  21093  prdstopn  21148  qtoptop  21220  imastopn  21240  vdgrfval  26164  suppss3  28682  ofcfval  29287  dya2iocuni  29472  trpredex  30827  stoweidlem27  38827  stoweidlem59  38859  omeiunle  39318
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