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Mirrors > Home > MPE Home > Th. List > wuncnv | Structured version Visualization version GIF version |
Description: A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wuncnv | ⊢ (𝜑 → ◡𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wunrn 9866 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
4 | 1, 2 | wundm 9865 | . . 3 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
5 | 1, 3, 4 | wunxp 9861 | . 2 ⊢ (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈) |
6 | cnvssrndm 5898 | . . 3 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)) |
8 | 1, 5, 7 | wunss 9849 | 1 ⊢ (𝜑 → ◡𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ⊆ wss 3798 × cxp 5340 ◡ccnv 5341 dom cdm 5342 ran crn 5343 WUnicwun 9837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-xp 5348 df-rel 5349 df-cnv 5350 df-dm 5352 df-rn 5353 df-wun 9839 |
This theorem is referenced by: wuntpos 9871 catcoppccl 17110 |
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