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

Theorem resfunexgALT 7881
Description: Alternate proof of resfunexg 7166, shorter but requiring ax-pow 5321 and ax-un 7673. (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 5962 . . . 4 (𝐵𝐶 → dom (𝐴𝐵) ∈ V)
21adantl 483 . . 3 ((Fun 𝐴𝐵𝐶) → dom (𝐴𝐵) ∈ V)
3 df-ima 5647 . . . 4 (𝐴𝐵) = ran (𝐴𝐵)
4 funimaexg 6588 . . . 4 ((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
53, 4eqeltrrid 2839 . . 3 ((Fun 𝐴𝐵𝐶) → ran (𝐴𝐵) ∈ V)
62, 5jca 513 . 2 ((Fun 𝐴𝐵𝐶) → (dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V))
7 xpexg 7685 . 2 ((dom (𝐴𝐵) ∈ V ∧ ran (𝐴𝐵) ∈ V) → (dom (𝐴𝐵) × ran (𝐴𝐵)) ∈ V)
8 relres 5967 . . . 4 Rel (𝐴𝐵)
9 relssdmrn 6221 . . . 4 (Rel (𝐴𝐵) → (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵)))
108, 9ax-mp 5 . . 3 (𝐴𝐵) ⊆ (dom (𝐴𝐵) × ran (𝐴𝐵))
11 ssexg 5281 . . 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 397  wcel 2107  Vcvv 3444  wss 3911   × cxp 5632  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  Rel wrel 5639  Fun wfun 6491
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 2704  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 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  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
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator