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

Proof of Theorem imaexALTV

Step	Hyp	Ref	Expression
1		imassrn 6087	. . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
2		rnexg 7920	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3		ssexg 5321	. . 3 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V)
4	1, 2, 3	sylancr 587	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
5		qsexg 8811	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V)
6		uniexg 7756	. . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V)
8		uniqsALTV 38308	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵))
9	8	eleq1d 2825	. . . 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 858	1 ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)

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