Theorem fnex 7213

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 7211. See fnexALT 7944 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 6635	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		df-fn 6537	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eleq1a 2864	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵))
4	3	impcom 412	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
5		resfunexg 7211	. . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
6	4, 5	sylan2 604	. . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
7	6	anassrs 472	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
8	2, 7	sylanb 592	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
9		resdm 6023	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
10	9	eleq1d 2854	. . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
11	10	biimpa 481	. 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
12	1, 8, 11	syl2an2r 697	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  dom cdm 5659   ↾ cres 5661  Rel wrel 5664  Fun wfun 6528   Fn wfn 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542

This theorem is referenced by:  fnexd  7214  funex  7215  fex  7222  offval  7681  fndmexb  7899  suppvalfn  8160  suppfnss  8181  fnsuppeq0  8184  fndmeng  9028  fdmfifsupp  9331  cfsmolem  10250  axcc2lem  10416  unirnfdomd  10548  prdsbas2  17518  prdsplusgval  17522  prdsmulrval  17524  prdsleval  17526  prdsdsval  17527  prdsvscaval  17528  xpscf  17615  brssc  17867  sscpwex  17868  ssclem  17872  isssc  17873  rescval2  17881  reschom  17883  isfuncd  17918  dprdw  20078  prdsmgp  20223  dsmmbas2  21852  dsmmelbas  21854  ptval  23692  prdstopn  23750  qtoptop  23822  imastopn  23842  fnpreimac  32952  suppss3  33005  ofcfval  34429  dya2iocuni  34614  tfsconcatun  43951  stoweidlem27  46628  stoweidlem59  46660  omeiunle  47118  preimafvelsetpreimafv  48021  fundcmpsurinjlem2  48032