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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 10758 | . . . . 5 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
4 | 1, 3 | wuncnv 10760 | . . . 4 ⊢ (𝜑 → ◡dom 𝐴 ∈ 𝑈) |
5 | 1 | wun0 10748 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈) |
6 | 1, 5 | wunsn 10746 | . . . 4 ⊢ (𝜑 → {∅} ∈ 𝑈) |
7 | 1, 4, 6 | wunun 10740 | . . 3 ⊢ (𝜑 → (◡dom 𝐴 ∪ {∅}) ∈ 𝑈) |
8 | 1, 2 | wunrn 10759 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
9 | 1, 7, 8 | wunxp 10754 | . 2 ⊢ (𝜑 → ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈) |
10 | tposssxp 8236 | . . 3 ⊢ tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴)) |
12 | 1, 9, 11 | wunss 10742 | 1 ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 ∅c0 4322 {csn 4630 × cxp 5676 ◡ccnv 5677 dom cdm 5678 ran crn 5679 tpos ctpos 8231 WUnicwun 10730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-tpos 8232 df-wun 10732 |
This theorem is referenced by: catcoppccl 18125 catcoppcclOLD 18126 |
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