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Theorem fnexALT 7641
 Description: Alternate proof of fnex 6962, derived using the Axiom of Replacement in the form of funimaexg 6413. This version uses ax-pow 5232 and ax-un 7448, whereas fnex 6962 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 6427 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 relssdmrn 6091 . . . 4 (Rel 𝐹𝐹 ⊆ (dom 𝐹 × ran 𝐹))
31, 2syl 17 . . 3 (𝐹 Fn 𝐴𝐹 ⊆ (dom 𝐹 × ran 𝐹))
43adantr 484 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
5 fndm 6428 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65eleq1d 2874 . . . 4 (𝐹 Fn 𝐴 → (dom 𝐹𝐵𝐴𝐵))
76biimpar 481 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → dom 𝐹𝐵)
8 fnfun 6426 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
9 funimaexg 6413 . . . . 5 ((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ∈ V)
108, 9sylan 583 . . . 4 ((𝐹 Fn 𝐴𝐴𝐵) → (𝐹𝐴) ∈ V)
11 imadmrn 5907 . . . . . . 7 (𝐹 “ dom 𝐹) = ran 𝐹
125imaeq2d 5897 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
1311, 12syl5eqr 2847 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
1413eleq1d 2874 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹𝐴) ∈ V))
1514biimpar 481 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝐹𝐴) ∈ V) → ran 𝐹 ∈ V)
1610, 15syldan 594 . . 3 ((𝐹 Fn 𝐴𝐴𝐵) → ran 𝐹 ∈ V)
17 xpexg 7460 . . 3 ((dom 𝐹𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V)
187, 16, 17syl2anc 587 . 2 ((𝐹 Fn 𝐴𝐴𝐵) → (dom 𝐹 × ran 𝐹) ∈ V)
19 ssexg 5192 . 2 ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V)
204, 18, 19syl2anc 587 1 ((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881   × cxp 5518  dom cdm 5520  ran crn 5521   “ cima 5523  Rel wrel 5525  Fun wfun 6321   Fn wfn 6322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-fun 6329  df-fn 6330 This theorem is referenced by: (None)
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