Theorem fnexALT 6313

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 5445. This version of fnex 5911 uses ax-pow 4292 and ax-un 4559, whereas fnex 5911 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 5459	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		relssdmrn 5288	. . . 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 5460	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	eleq1d 2303	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	6	biimpar 297	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
8		fnfun 5458	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
9		funimaexg 5445	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
10	8, 9	sylan 283	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
11		imadmrn 5116	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
12	5	imaeq2d 5106	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
13	11, 12	eqtr3id 2281	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
14	13	eleq1d 2303	. . . . 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 4869	. . 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 4254	. 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 2205  Vcvv 2815   ⊆ wss 3214   × cxp 4752  dom cdm 4754  ran crn 4755   “ cima 4757  Rel wrel 4759  Fun wfun 5351   Fn wfn 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359  df-fn 5360

This theorem is referenced by: (None)