Theorem fnex 6755

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 6753. See fnexALT 7413 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 6236	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		df-fn 6140	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
3		eleq1a 2854	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵))
4	3	impcom 398	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
5		resfunexg 6753	. . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
6	4, 5	sylan2 586	. . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
7	6	anassrs 461	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
8	2, 7	sylanb 576	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
9		resdm 5693	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
10	9	eleq1d 2844	. . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
11	10	biimpa 470	. 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
12	1, 8, 11	syl2an2r 675	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  dom cdm 5357   ↾ cres 5359  Rel wrel 5362  Fun wfun 6131   Fn wfn 6132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145

This theorem is referenced by:  funex  6756  fex  6763  offval  7183  ofrfval  7184  suppvalfn  7585  suppfnss  7603  suppfnssOLD  7604  fnsuppeq0  7607  wfrlem15  7714  fndmeng  8321  fdmfifsupp  8575  cfsmolem  9429  axcc2lem  9595  unirnfdomd  9726  prdsbas2  16526  prdsplusgval  16530  prdsmulrval  16532  prdsleval  16534  prdsdsval  16535  prdsvscaval  16536  brssc  16870  sscpwex  16871  ssclem  16875  isssc  16876  rescval2  16884  reschom  16886  rescabs  16889  isfuncd  16921  dprdw  18807  prdsmgp  19008  dsmmbas2  20491  dsmmelbas  20493  ptval  21793  elptr  21796  prdstopn  21851  qtoptop  21923  imastopn  21943  fnpreimac  30053  suppss3  30085  ofcfval  30766  dya2iocuni  30951  trpredex  32333  fnexd  40263  stoweidlem27  41185  stoweidlem59  41217  omeiunle  41672