Theorem fnexALT 7909

Description: Alternate proof of fnex 7173, derived using the Axiom of Replacement in the form of funimaexg 6587. This version uses ax-pow 5315 and ax-un 7691, whereas fnex 7173 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 6602	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		relssdmrn 6229	. . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4	3	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5		fndm 6603	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	eleq1d 2813	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7	6	biimpar 477	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵)
8		fnfun 6600	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
9		funimaexg 6587	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
10	8, 9	sylan 580	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V)
11		imadmrn 6030	. . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
12	5	imaeq2d 6020	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
13	11, 12	eqtr3id 2778	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
14	13	eleq1d 2813	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹 “ 𝐴) ∈ V))
15	14	biimpar 477	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 “ 𝐴) ∈ V) → ran 𝐹 ∈ V)
16	10, 15	syldan 591	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → ran 𝐹 ∈ V)
17		xpexg 7706	. . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
18	7, 16, 17	syl2anc 584	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19		ssexg 5273	. 2 ⊢ ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
20	4, 18, 19	syl2anc 584	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911   × cxp 5629  dom cdm 5631  ran crn 5632   “ cima 5634  Rel wrel 5636  Fun wfun 6493   Fn wfn 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502

This theorem is referenced by: (None)