Theorem fnex 2740

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 funimaexg 2715.

Assertion

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

Proof of Theorem fnex

Step	Hyp	Ref	Expression
1		fndm 2723	. . . . . . 7 ⊢ (F Fn A → dom F = A)
2	1	eleq1d 1155	. . . . . 6 ⊢ (F Fn A → (dom F ∈ B ↔ A ∈ B))
3	2	biimprd 136	. . . . 5 ⊢ (F Fn A → (A ∈ B → dom F ∈ B))
4		funimaexg 2715	. . . . . . . 8 ⊢ (A ∈ B → (Fun F → (F “ A) ∈ V))
5		fnfun 2721	. . . . . . . 8 ⊢ (F Fn A → Fun F)
6	4, 5	syl5 22	. . . . . . 7 ⊢ (A ∈ B → (F Fn A → (F “ A) ∈ V))
7	6	com12 13	. . . . . 6 ⊢ (F Fn A → (A ∈ B → (F “ A) ∈ V))
8		imaeq2 2603	. . . . . . . . 9 ⊢ (dom F = A → (F “ dom F) = (F “ A))
9	1, 8	syl 12	. . . . . . . 8 ⊢ (F Fn A → (F “ dom F) = (F “ A))
10		imadmrn 2610	. . . . . . . 8 ⊢ (F “ dom F) = ran F
11	9, 10	syl5eqr 1138	. . . . . . 7 ⊢ (F Fn A → ran F = (F “ A))
12	11	eleq1d 1155	. . . . . 6 ⊢ (F Fn A → (ran F ∈ V ↔ (F “ A) ∈ V))
13	7, 12	sylibrd 179	. . . . 5 ⊢ (F Fn A → (A ∈ B → ran F ∈ V))
14	3, 13	jcad 455	. . . 4 ⊢ (F Fn A → (A ∈ B → (dom F ∈ B ∧ ran F ∈ V)))
15		xpexg 2489	. . . 4 ⊢ ((dom F ∈ B ∧ ran F ∈ V) → (dom F × ran F) ∈ V)
16	14, 15	syl6 23	. . 3 ⊢ (F Fn A → (A ∈ B → (dom F × ran F) ∈ V))
17		ssexg 1702	. . . . 5 ⊢ ((dom F × ran F) ∈ V → (F ⊆ (dom F × ran F) → F ∈ V))
18		fnrel 2722	. . . . . 6 ⊢ (F Fn A → Rel F)
19		relssdr 2668	. . . . . 6 ⊢ (Rel F → F ⊆ (dom F × ran F))
20	18, 19	syl 12	. . . . 5 ⊢ (F Fn A → F ⊆ (dom F × ran F))
21	17, 20	syl5 22	. . . 4 ⊢ ((dom F × ran F) ∈ V → (F Fn A → F ∈ V))
22	21	com12 13	. . 3 ⊢ (F Fn A → ((dom F × ran F) ∈ V → F ∈ V))
23	16, 22	syld 27	. 2 ⊢ (F Fn A → (A ∈ B → F ∈ V))
24	23	com12 13	1 ⊢ (A ∈ B → (F Fn A → F ∈ V))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   × cxp 2408  dom cdm 2410  ran crn 2411   “ cima 2413  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  funex 2741  fex 2771  tfrlem12 2960  f1oeng 3298  f1domg 3299  unfilem3 3440  aceq3lem 3555  ac6lem 3575  clim2 4881  hcauchy 5103  hlim2 5112  chlim 5139  hosmvalt 5487  hodmvalt 5488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433

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