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Theorem imaexALTV 38273
Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7930) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38279. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)
Assertion
Ref Expression
imaexALTV ((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)

Proof of Theorem imaexALTV
StepHypRef Expression
1 imassrn 6085 . . 3 (𝐴𝐵) ⊆ ran 𝐴
2 rnexg 7919 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
3 ssexg 5324 . . 3 (((𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3sylancr 586 . 2 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 qsexg 8808 . . . . 5 (𝐵𝑋 → (𝐵 / 𝐴) ∈ V)
6 uniexg 7752 . . . . 5 ((𝐵 / 𝐴) ∈ V → (𝐵 / 𝐴) ∈ V)
75, 6syl 17 . . . 4 (𝐵𝑋 (𝐵 / 𝐴) ∈ V)
8 uniqsALTV 38272 . . . . 5 ((𝐴𝐵) ∈ 𝑊 (𝐵 / 𝐴) = (𝐴𝐵))
98eleq1d 2822 . . . 4 ((𝐴𝐵) ∈ 𝑊 → ( (𝐵 / 𝐴) ∈ V ↔ (𝐴𝐵) ∈ V))
107, 9imbitrid 244 . . 3 ((𝐴𝐵) ∈ 𝑊 → (𝐵𝑋 → (𝐴𝐵) ∈ V))
1110imp 406 . 2 (((𝐴𝐵) ∈ 𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
124, 11jaoi 856 1 ((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  wcel 2104  Vcvv 3477  wss 3963   cuni 4914  ran crn 5684  cres 5685  cima 5686   / cqs 8737
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5689  df-rel 5690  df-cnv 5691  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8740  df-qs 8744
This theorem is referenced by:  ecexALTV  38274  cnvepimaex  38279
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