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| Mirrors > Home > MPE Home > Th. List > wuntpos | Structured version Visualization version GIF version | ||
| Description: A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) | 
| wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) | 
| Ref | Expression | 
|---|---|
| wuntpos | ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 2 | wunop.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 3 | 1, 2 | wundm 10768 | . . . . 5 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) | 
| 4 | 1, 3 | wuncnv 10770 | . . . 4 ⊢ (𝜑 → ◡dom 𝐴 ∈ 𝑈) | 
| 5 | 1 | wun0 10758 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈) | 
| 6 | 1, 5 | wunsn 10756 | . . . 4 ⊢ (𝜑 → {∅} ∈ 𝑈) | 
| 7 | 1, 4, 6 | wunun 10750 | . . 3 ⊢ (𝜑 → (◡dom 𝐴 ∪ {∅}) ∈ 𝑈) | 
| 8 | 1, 2 | wunrn 10769 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) | 
| 9 | 1, 7, 8 | wunxp 10764 | . 2 ⊢ (𝜑 → ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈) | 
| 10 | tposssxp 8255 | . . 3 ⊢ tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) | |
| 11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴)) | 
| 12 | 1, 9, 11 | wunss 10752 | 1 ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 {csn 4626 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 tpos ctpos 8250 WUnicwun 10740 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-tpos 8251 df-wun 10742 | 
| This theorem is referenced by: catcoppccl 18162 | 
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