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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 7930) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38279. (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 6085 | . . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 | |
2 | rnexg 7919 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
3 | ssexg 5324 | . . 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 8808 | . . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V) | |
6 | uniexg 7752 | . . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V) |
8 | uniqsALTV 38272 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵)) | |
9 | 8 | eleq1d 2822 | . . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 ∈ wcel 2104 Vcvv 3477 ⊆ wss 3963 ∪ cuni 4914 ran crn 5684 ↾ cres 5685 “ cima 5686 / cqs 8737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-xp 5689 df-rel 5690 df-cnv 5691 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ec 8740 df-qs 8744 |
This theorem is referenced by: ecexALTV 38274 cnvepimaex 38279 |
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