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Theorem tposssxp 8271
Description: The transposition is a subset of a Cartesian product. (Contributed by Mario Carneiro, 12-Jan-2017.)
Assertion
Ref Expression
tposssxp tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)

Proof of Theorem tposssxp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef 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)
31, 2eqsstri 4043 . 2 tpos 𝐹 ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
4 eqid 2740 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
54dmmptss 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))
75, 6ax-mp 5 . 2 (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹) ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
83, 7sstri 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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