Theorem fnex 5871

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 5870. (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 5425	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2	1	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → Rel 𝐹)
3		df-fn 5327	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
4		eleq1a 2301	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (dom 𝐹 = 𝐴 → dom 𝐹 ∈ 𝐵))
5	4	impcom 125	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
6		resfunexg 5870	. . . . 5 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
7	5, 6	sylan2 286	. . . 4 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 = 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐹 ↾ dom 𝐹) ∈ V)
8	7	anassrs 400	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
9	3, 8	sylanb 284	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 ↾ dom 𝐹) ∈ V)
10		resdm 5050	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
11	10	eleq1d 2298	. . 3 ⊢ (Rel 𝐹 → ((𝐹 ↾ dom 𝐹) ∈ V ↔ 𝐹 ∈ V))
12	11	biimpa 296	. 2 ⊢ ((Rel 𝐹 ∧ (𝐹 ↾ dom 𝐹) ∈ V) → 𝐹 ∈ V)
13	2, 9, 12	syl2anc 411	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800  dom cdm 4723   ↾ cres 4725  Rel wrel 4728  Fun wfun 5318   Fn wfn 5319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332

This theorem is referenced by:  funex  5872  fex  5878  offval  6238  ofrfval  6239  uchoice  6295  tfrlemibex  6490  tfr1onlembex  6506  fndmeng  6980  cc2lem  7475  frecfzennn  10678  prdsbas2  13352  prdsplusgval  13356  prdsplusgfval  13357  prdsmulrval  13358  prdsmulrfval  13359  xpscf  13420  mulgval  13699  mulgfng  13701  invrfvald  14126