MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnexALT Structured version   Visualization version   GIF version

Theorem fnexALT 7976
Description: Alternate proof of fnex 7238, derived using the Axiom of Replacement in the form of funimaexg 6652. This version uses ax-pow 5364 and ax-un 7756, whereas fnex 7238 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 6669 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 relssdmrn 6287 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . . 3 (𝐹 Fn 𝐴𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 480 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 6670 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65eleq1d 2825 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵𝐴𝐵))
76biimpar 477 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
8 fnfun 6667 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
9 funimaexg 6652 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
108, 9sylan 580 . . . 4 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐴) ∈ V)
11 imadmrn 6087 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 6077 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
1311, 12eqtr3id 2790 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
1413eleq1d 2825 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹𝐴) ∈ V))
1514biimpar 477 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝐴) ∈ V) → ran 𝐹 ∈ V)
1610, 15syldan 591 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → ran 𝐹 ∈ V)
17 xpexg 7771 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
187, 16, 17syl2anc 584 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19 ssexg 5322 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
204, 18, 19syl2anc 584 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  Vcvv 3479  wss 3950   × cxp 5682  dom cdm 5684  ran crn 5685  cima 5687  Rel wrel 5689  Fun wfun 6554   Fn wfn 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator