Theorem imaexALTV 38278

Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7947) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38284. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.)

Assertion

Ref	Expression
imaexALTV	⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem imaexALTV

Step	Hyp	Ref	Expression
1		imassrn 6095	. . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
2		rnexg 7936	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3		ssexg 5341	. . 3 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V)
4	1, 2, 3	sylancr 586	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
5		qsexg 8827	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V)
6		uniexg 7769	. . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V)
8		uniqsALTV 38277	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵))
9	8	eleq1d 2829	. . . 4 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (∪ (𝐵 / 𝐴) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
10	7, 9	imbitrid 244	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (𝐵 ∈ 𝑋 → (𝐴 “ 𝐵) ∈ V))
11	10	imp 406	. 2 ⊢ (((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 “ 𝐵) ∈ V)
12	4, 11	jaoi 856	1 ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∪ cuni 4931  ran crn 5696   ↾ cres 5697   “ cima 5698   / cqs 8756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-xp 5701  df-rel 5702  df-cnv 5703  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-ec 8759  df-qs 8763

This theorem is referenced by:  ecexALTV  38279  cnvepimaex  38284