Theorem fnex 6436

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 6434. See fnexALT 7082 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 5949	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	adantr 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → Rel 𝐹)
3		df-fn 5853	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4		eleq1a 2699	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵))
5	4	impcom 446	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
6		resfunexg 6434	. . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
7	5, 6	sylan2 491	. . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
8	7	anassrs 679	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
9	3, 8	sylanb 489	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10		resdm 5404	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
11	10	eleq1d 2688	. . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
12	11	biimpa 501	. 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
13	2, 9, 12	syl2anc 692	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  Vcvv 3191  dom cdm 5079   ↾ cres 5081  Rel wrel 5084  Fun wfun 5844   Fn wfn 5845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858

This theorem is referenced by:  funex  6437  fex  6445  offval  6858  ofrfval  6859  suppvalfn  7248  suppfnss  7266  fnsuppeq0  7269  wfrlem15  7375  fndmeng  7979  fdmfifsupp  8230  cfsmolem  9037  axcc2lem  9203  unirnfdomd  9334  prdsbas2  16045  prdsplusgval  16049  prdsmulrval  16051  prdsleval  16053  prdsdsval  16054  prdsvscaval  16055  brssc  16390  sscpwex  16391  ssclem  16395  isssc  16396  rescval2  16404  reschom  16406  rescabs  16409  isfuncd  16441  dprdw  18325  prdsmgp  18526  dsmmbas2  19995  dsmmelbas  19997  ptval  21278  elptr  21281  prdstopn  21336  qtoptop  21408  imastopn  21428  suppss3  29336  ofcfval  29933  dya2iocuni  30118  trpredex  31426  fnexd  38780  stoweidlem27  39519  stoweidlem59  39551  omeiunle  40006