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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 8249 | . . 3 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) | |
2 | cossxp 6293 | . . 3 ⊢ (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) | |
3 | 1, 2 | eqsstri 4029 | . 2 ⊢ tpos 𝐹 ⊆ (dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) × ran 𝐹) |
4 | eqid 2734 | . . . 4 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) | |
5 | 4 | dmmptss 6262 | . . 3 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ⊆ (◡dom 𝐹 ∪ {∅}) |
6 | xpss1 5707 | . . 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 4004 | 1 ⊢ tpos 𝐹 ⊆ ((◡dom 𝐹 ∪ {∅}) × ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3960 ⊆ wss 3962 ∅c0 4338 {csn 4630 ∪ cuni 4911 ↦ cmpt 5230 × cxp 5686 ◡ccnv 5687 dom cdm 5688 ran crn 5689 ∘ ccom 5692 tpos ctpos 8248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-mpt 5231 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-tpos 8249 |
This theorem is referenced by: reltpos 8254 tposexg 8263 wuntpos 10771 catcoppccl 18170 catcoppcclOLD 18171 |
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