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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 7671) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 36157. (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 5925 | . . 3 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 | |
2 | rnexg 7660 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
3 | ssexg 5201 | . . 3 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V) | |
4 | 1, 2, 3 | sylancr 590 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V) |
5 | qsexg 8435 | . . . . 5 ⊢ (𝐵 ∈ 𝑋 → (𝐵 / 𝐴) ∈ V) | |
6 | uniexg 7506 | . . . . 5 ⊢ ((𝐵 / 𝐴) ∈ V → ∪ (𝐵 / 𝐴) ∈ V) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐵 ∈ 𝑋 → ∪ (𝐵 / 𝐴) ∈ V) |
8 | uniqsALTV 36150 | . . . . 5 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → ∪ (𝐵 / 𝐴) = (𝐴 “ 𝐵)) | |
9 | 8 | eleq1d 2815 | . . . 4 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (∪ (𝐵 / 𝐴) ∈ V ↔ (𝐴 “ 𝐵) ∈ V)) |
10 | 7, 9 | syl5ib 247 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ∈ 𝑊 → (𝐵 ∈ 𝑋 → (𝐴 “ 𝐵) ∈ V)) |
11 | 10 | imp 410 | . 2 ⊢ (((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 “ 𝐵) ∈ V) |
12 | 4, 11 | jaoi 857 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 ∪ cuni 4805 ran crn 5537 ↾ cres 5538 “ cima 5539 / cqs 8368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ec 8371 df-qs 8375 |
This theorem is referenced by: ecexALTV 36152 cnvepimaex 36157 |
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