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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 10139 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
4 | 1, 2 | wundm 10138 | . . 3 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
5 | 1, 3, 4 | wunxp 10134 | . 2 ⊢ (𝜑 → (ran 𝐴 × dom 𝐴) ∈ 𝑈) |
6 | cnvssrndm 6115 | . . 3 ⊢ ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → ◡𝐴 ⊆ (ran 𝐴 × dom 𝐴)) |
8 | 1, 5, 7 | wunss 10122 | 1 ⊢ (𝜑 → ◡𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 × cxp 5546 ◡ccnv 5547 dom cdm 5548 ran crn 5549 WUnicwun 10110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-wun 10112 |
This theorem is referenced by: wuntpos 10144 catcoppccl 17356 |
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