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Theorem imaexALTV 38309
Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7931) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38315. (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 6087 . . 3 (𝐴𝐵) ⊆ ran 𝐴
2 rnexg 7920 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
3 ssexg 5321 . . 3 (((𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3sylancr 587 . 2 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 qsexg 8811 . . . . 5 (𝐵𝑋 → (𝐵 / 𝐴) ∈ V)
6 uniexg 7756 . . . . 5 ((𝐵 / 𝐴) ∈ V → (𝐵 / 𝐴) ∈ V)
75, 6syl 17 . . . 4 (𝐵𝑋 (𝐵 / 𝐴) ∈ V)
8 uniqsALTV 38308 . . . . 5 ((𝐴𝐵) ∈ 𝑊 (𝐵 / 𝐴) = (𝐴𝐵))
98eleq1d 2825 . . . 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 858 1 ((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  wcel 2108  Vcvv 3479  wss 3950   cuni 4905  ran crn 5684  cres 5685  cima 5686   / cqs 8740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  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-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-xp 5689  df-rel 5690  df-cnv 5691  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8743  df-qs 8747
This theorem is referenced by:  ecexALTV  38310  cnvepimaex  38315
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