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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 7898 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | dmexg 7615 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
3 | cnvexg 7631 | . . . . 5 ⊢ (dom 𝐹 ∈ V → ◡dom 𝐹 ∈ V) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ◡dom 𝐹 ∈ V) |
5 | p0ex 5287 | . . . 4 ⊢ {∅} ∈ V | |
6 | unexg 7474 | . . . 4 ⊢ ((◡dom 𝐹 ∈ V ∧ {∅} ∈ V) → (◡dom 𝐹 ∪ {∅}) ∈ V) | |
7 | 4, 5, 6 | sylancl 588 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (◡dom 𝐹 ∪ {∅}) ∈ V) |
8 | rnexg 7616 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
9 | 7, 8 | xpexd 7476 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) |
10 | ssexg 5229 | . 2 ⊢ ((tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V) | |
11 | 1, 9, 10 | sylancr 589 | 1 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 ∅c0 4293 {csn 4569 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 tpos ctpos 7893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-tpos 7894 |
This theorem is referenced by: tposex 7928 oftpos 21063 |
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