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Theorem resfunexgALT 7943
Description: Alternate proof of resfunexg 7221, shorter but requiring ax-pow 5359 and ax-un 7732. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resfunexgALT ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)

Proof of Theorem resfunexgALT
StepHypRef Expression
1 dmresexg 6003 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
21adantl 481 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
3 df-ima 5685 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
4 funimaexg 6633 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
53, 4eqeltrrid 2833 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
62, 5jca 511 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V))
7 xpexg 7744 . 2 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
8 relres 6008 . . . 4 Rel (𝐴𝐵)
9 relssdmrn 6266 . . . 4 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
108, 9ax-mp 5 . . 3 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
11 ssexg 5317 . . 3 (((𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∧ (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
1210, 11mpan 689 . 2 ((dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V → (𝐴𝐵) ∈ V)
136, 7, 123syl 18 1 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  Vcvv 3469  wss 3944   × cxp 5670  dom cdm 5672  ran crn 5673  cres 5674  cima 5675  Rel wrel 5677  Fun wfun 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544
This theorem is referenced by: (None)
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