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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 10144 | . . . . 5 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
4 | 1, 3 | wuncnv 10146 | . . . 4 ⊢ (𝜑 → ◡dom 𝐴 ∈ 𝑈) |
5 | 1 | wun0 10134 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈) |
6 | 1, 5 | wunsn 10132 | . . . 4 ⊢ (𝜑 → {∅} ∈ 𝑈) |
7 | 1, 4, 6 | wunun 10126 | . . 3 ⊢ (𝜑 → (◡dom 𝐴 ∪ {∅}) ∈ 𝑈) |
8 | 1, 2 | wunrn 10145 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
9 | 1, 7, 8 | wunxp 10140 | . 2 ⊢ (𝜑 → ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈) |
10 | tposssxp 7890 | . . 3 ⊢ tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴)) |
12 | 1, 9, 11 | wunss 10128 | 1 ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 {csn 4561 × cxp 5548 ◡ccnv 5549 dom cdm 5550 ran crn 5551 tpos ctpos 7885 WUnicwun 10116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-tpos 7886 df-wun 10118 |
This theorem is referenced by: catcoppccl 17362 |
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