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Mirrors > Home > ILE Home > Th. List > tposexg | 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 6243 | . 2 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) | |
2 | dmexg 4886 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
3 | cnvexg 5161 | . . . . 5 ⊢ (dom 𝐹 ∈ V → ◡dom 𝐹 ∈ V) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → ◡dom 𝐹 ∈ V) |
5 | p0ex 4185 | . . . 4 ⊢ {∅} ∈ V | |
6 | unexg 4439 | . . . 4 ⊢ ((◡dom 𝐹 ∈ V ∧ {∅} ∈ V) → (◡dom 𝐹 ∪ {∅}) ∈ V) | |
7 | 4, 5, 6 | sylancl 413 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (◡dom 𝐹 ∪ {∅}) ∈ V) |
8 | rnexg 4887 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
9 | xpexg 4736 | . . 3 ⊢ (((◡dom 𝐹 ∪ {∅}) ∈ V ∧ ran 𝐹 ∈ V) → ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) | |
10 | 7, 8, 9 | syl2anc 411 | . 2 ⊢ (𝐹 ∈ 𝑉 → ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) |
11 | ssexg 4139 | . 2 ⊢ ((tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V) | |
12 | 1, 10, 11 | sylancr 414 | 1 ⊢ (𝐹 ∈ 𝑉 → tpos 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 ⊆ wss 3129 ∅c0 3422 {csn 3591 × cxp 4620 ◡ccnv 4621 dom cdm 4622 ran crn 4623 tpos ctpos 6238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-tpos 6239 |
This theorem is referenced by: tposex 6272 opprvalg 13053 opprmulfvalg 13054 opprex 13057 opprsllem 13058 |
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