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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cossxrncnvepresex | Structured version Visualization version GIF version |
Description: Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
1cossxrncnvepresex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrncnvepresex 38387 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | |
2 | cossex 38398 | . 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3479 E cep 5581 ◡ccnv 5682 ↾ cres 5685 ⋉ cxrn 38159 ≀ ccoss 38160 |
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-pow 5363 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-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 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-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-eprel 5582 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-fo 6565 df-fv 6567 df-1st 8010 df-2nd 8011 df-xrn 38350 df-coss 38390 |
This theorem is referenced by: pets 38831 |
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