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Theorem imaexALTV 35746
Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7606) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 35752. (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 5911 . . 3 (𝐴𝐵) ⊆ ran 𝐴
2 rnexg 7599 . . 3 (𝐴𝑉 → ran 𝐴 ∈ V)
3 ssexg 5194 . . 3 (((𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3sylancr 590 . 2 (𝐴𝑉 → (𝐴𝐵) ∈ V)
5 qsexg 8342 . . . . 5 (𝐵𝑋 → (𝐵 / 𝐴) ∈ V)
6 uniexg 7450 . . . . 5 ((𝐵 / 𝐴) ∈ V → (𝐵 / 𝐴) ∈ V)
75, 6syl 17 . . . 4 (𝐵𝑋 (𝐵 / 𝐴) ∈ V)
8 uniqsALTV 35745 . . . . 5 ((𝐴𝐵) ∈ 𝑊 (𝐵 / 𝐴) = (𝐴𝐵))
98eleq1d 2877 . . . 4 ((𝐴𝐵) ∈ 𝑊 → ( (𝐵 / 𝐴) ∈ V ↔ (𝐴𝐵) ∈ V))
107, 9syl5ib 247 . . 3 ((𝐴𝐵) ∈ 𝑊 → (𝐵𝑋 → (𝐴𝐵) ∈ V))
1110imp 410 . 2 (((𝐴𝐵) ∈ 𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
124, 11jaoi 854 1 ((𝐴𝑉 ∨ ((𝐴𝐵) ∈ 𝑊𝐵𝑋)) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  wcel 2112  Vcvv 3444  wss 3884   cuni 4803  ran crn 5524  cres 5525  cima 5526   / cqs 8275
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ec 8278  df-qs 8282
This theorem is referenced by:  ecexALTV  35747  cnvepimaex  35752
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