Theorem fnexALT 6165

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 5339. This version of fnex 5781 uses ax-pow 4204 and ax-un 4465, whereas fnex 5781 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem fnexALT

Step	Hyp	Ref	Expression
1		fnrel 5353	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		relssdmrn 5187	. . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 14	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4	3	adantr 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5		fndm 5354	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	eleq1d 2262	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	6	biimpar 297	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
8		fnfun 5352	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
9		funimaexg 5339	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
10	8, 9	sylan 283	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
11		imadmrn 5016	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
12	5	imaeq2d 5006	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
13	11, 12	eqtr3id 2240	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
14	13	eleq1d 2262	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹 “ 𝐴) ∈ V))
15	14	biimpar 297	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 “ 𝐴) ∈ V) → ran 𝐹 ∈ V)
16	10, 15	syldan 282	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → ran 𝐹 ∈ V)
17		xpexg 4774	. . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
18	7, 16, 17	syl2anc 411	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19		ssexg 4169	. 2 ⊢ ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
20	4, 18, 19	syl2anc 411	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154   × cxp 4658  dom cdm 4660  ran crn 4661   “ cima 4663  Rel wrel 4665  Fun wfun 5249   Fn wfn 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-fun 5257  df-fn 5258

This theorem is referenced by: (None)