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Mirrors > Home > MPE Home > Th. List > Mathboxes > imaexALTV | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
imaexALTV | ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V) |
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) |
Colors of variables: wff setvar class |
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 |
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