Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9550 | . . . . 5 ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) |
4 | 1, 3 | wuncnv 9552 | . . . 4 ⊢ (𝜑 → ◡dom 𝐴 ∈ 𝑈) |
5 | 1 | wun0 9540 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈) |
6 | 1, 5 | wunsn 9538 | . . . 4 ⊢ (𝜑 → {∅} ∈ 𝑈) |
7 | 1, 4, 6 | wunun 9532 | . . 3 ⊢ (𝜑 → (◡dom 𝐴 ∪ {∅}) ∈ 𝑈) |
8 | 1, 2 | wunrn 9551 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
9 | 1, 7, 8 | wunxp 9546 | . 2 ⊢ (𝜑 → ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) ∈ 𝑈) |
10 | tposssxp 7356 | . . 3 ⊢ tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴) | |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → tpos 𝐴 ⊆ ((◡dom 𝐴 ∪ {∅}) × ran 𝐴)) |
12 | 1, 9, 11 | wunss 9534 | 1 ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 {csn 4177 × cxp 5112 ◡ccnv 5113 dom cdm 5114 ran crn 5115 tpos ctpos 7351 WUnicwun 9522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-tpos 7352 df-wun 9524 |
This theorem is referenced by: catcoppccl 16758 |
Copyright terms: Public domain | W3C validator |