Theorem imaexALTV 36465

Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7762) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 36471. (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 5980	. . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
2		rnexg 7751	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3		ssexg 5247	. . 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 8564	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V)
6		uniexg 7593	. . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V)
8		uniqsALTV 36464	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵))
9	8	eleq1d 2823	. . . 4 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (∪ (𝐵 / 𝐴) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
10	7, 9	syl5ib 243	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (𝐵 ∈ 𝑋 → (𝐴 “ 𝐵) ∈ V))
11	10	imp 407	. 2 ⊢ (((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 “ 𝐵) ∈ V)
12	4, 11	jaoi 854	1 ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ran crn 5590   ↾ cres 5591   “ cima 5592   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  ecexALTV  36466  cnvepimaex  36471