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Mirrors > Home > MPE Home > Th. List > tposssxp | Structured version Visualization version GIF version |
Description: The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
tposssxp | ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tpos 8267 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
2 | cossxp 6303 | . . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) | |
3 | 1, 2 | eqsstri 4043 | . 2 ⊢ tpos 𝐹 ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) |
4 | eqid 2740 | . . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
5 | 4 | dmmptss 6272 | . . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) |
6 | xpss1 5719 | . . 3 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) → (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹)) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
8 | 3, 7 | sstri 4018 | 1 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3974 ⊆ wss 3976 ∅c0 4352 {csn 4648 ∪ cuni 4931 ↦ cmpt 5249 × cxp 5698 ◡ccnv 5699 dom cdm 5700 ran crn 5701 ∘ ccom 5704 tpos ctpos 8266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-mpt 5250 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-tpos 8267 |
This theorem is referenced by: reltpos 8272 tposexg 8281 wuntpos 10803 catcoppccl 18184 catcoppcclOLD 18185 |
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