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Mirrors > Home > MPE Home > Th. List > tposexg | Structured version Visualization version GIF version |
Description: The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposexg | ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposssxp 8240 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | dmexg 7913 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
3 | cnvexg 7936 | . . . . 5 ⊢ (dom 𝐹 ∈ V → ◡dom 𝐹 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ◡dom 𝐹 ∈ V) |
5 | p0ex 5386 | . . . 4 ⊢ {∅} ∈ V | |
6 | unexg 7755 | . . . 4 ⊢ ((◡dom 𝐹 ∈ V ∧ {∅} ∈ V) → (◡dom 𝐹 ∪ {∅}) ∈ V) | |
7 | 4, 5, 6 | sylancl 584 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (◡dom 𝐹 ∪ {∅}) ∈ V) |
8 | rnexg 7914 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
9 | 7, 8 | xpexd 7757 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) |
10 | ssexg 5325 | . 2 ⊢ ((tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V) | |
11 | 1, 9, 10 | sylancr 585 | 1 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3471 ∪ cun 3945 ⊆ wss 3947 ∅c0 4324 {csn 4630 × cxp 5678 ◡ccnv 5679 dom cdm 5680 ran crn 5681 tpos ctpos 8235 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-tpos 8236 |
This theorem is referenced by: tposex 8270 oftpos 22372 |
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