Theorem imaexALTV 35602

Description: Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7620) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 35608. (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 5940	. . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
2		rnexg 7614	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3		ssexg 5227	. . 3 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V)
4	1, 2, 3	sylancr 589	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)
5		qsexg 8355	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V)
6		uniexg 7466	. . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V)
7	5, 6	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V)
8		uniqsALTV 35601	. . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵))
9	8	eleq1d 2897	. . . 4 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (∪ (𝐵 / 𝐴) ∈ V ↔ (𝐴 “ 𝐵) ∈ V))
10	7, 9	syl5ib 246	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (𝐵 ∈ 𝑋 → (𝐴 “ 𝐵) ∈ V))
11	10	imp 409	. 2 ⊢ (((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 “ 𝐵) ∈ V)
12	4, 11	jaoi 853	1 ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  ∪ cuni 4838  ran crn 5556   ↾ cres 5557   “ cima 5558   / cqs 8288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ec 8291  df-qs 8295

This theorem is referenced by:  ecexALTV  35603  cnvepimaex  35608