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Theorem fnexALT 7884
Description: Alternate proof of fnex 7168, derived using the Axiom of Replacement in the form of funimaexg 6588. This version uses ax-pow 5321 and ax-un 7673, whereas fnex 7168 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fnexALT ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Proof of Theorem fnexALT
StepHypRef Expression
1 fnrel 6605 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 relssdmrn 6221 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . . 3 (𝐹 Fn 𝐴𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 482 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 6606 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65eleq1d 2823 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵𝐴𝐵))
76biimpar 479 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
8 fnfun 6603 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
9 funimaexg 6588 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
108, 9sylan 581 . . . 4 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐴) ∈ V)
11 imadmrn 6024 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 6014 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
1311, 12eqtr3id 2791 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
1413eleq1d 2823 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹𝐴) ∈ V))
1514biimpar 479 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝐴) ∈ V) → ran 𝐹 ∈ V)
1610, 15syldan 592 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → ran 𝐹 ∈ V)
17 xpexg 7685 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
187, 16, 17syl2anc 585 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19 ssexg 5281 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
204, 18, 19syl2anc 585 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3446  wss 3911   × cxp 5632  dom cdm 5634  ran crn 5635  cima 5637  Rel wrel 5639  Fun wfun 6491   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500
This theorem is referenced by: (None)
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