Theorem fnex 7254

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 7252. See fnexALT 7997 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fnex	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Proof of Theorem fnex

Step	Hyp	Ref	Expression
1		fnrel 6671	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		df-fn 6569	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eleq1a 2859	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵))
4	3	impcom 413	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
5		resfunexg 7252	. . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
6	4, 5	sylan2 605	. . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
7	6	anassrs 473	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
8	2, 7	sylanb 593	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
9		resdm 6033	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
10	9	eleq1d 2849	. . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
11	10	biimpa 482	. 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
12	1, 8, 11	syl2an2r 698	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  dom cdm 5658   ↾ cres 5660  Rel wrel 5663  Fun wfun 6560   Fn wfn 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574

This theorem is used by:  fnexd  7255  funex  7256  fex  7263  offval  7734  fndmexb  7952  suppvalfn  8223  suppfnss  8244  fnsuppeq0  8247  fndmeng  9101  fdmfifsupp  9404  cfsmolem  10387  axcc2lem  10553  unirnfdomd  10691  prdsbas2  17690  prdsplusgval  17694  prdsmulrval  17696  prdsleval  17698  prdsdsval  17699  prdsvscaval  17700  xpscf  17787  brssc  18039  sscpwex  18040  ssclem  18044  isssc  18045  rescval2  18053  reschom  18055  isfuncd  18090  dprdw  20277  prdsmgp  20422  dsmmbas2  22098  dsmmelbas  22100  ptval  23944  prdstopn  24002  qtoptop  24074  imastopn  24094  fnpreimac  33362  suppss3  33411  ofcfval  34841  dya2iocuni  35026  tfsconcatun  44600  stoweidlem27  47310  stoweidlem59  47342  omeiunle  47800  preimafvelsetpreimafv  48743  fundcmpsurinjlem2  48754  elixpmpt  50335