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Theorem tposssxp 8255
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 8251 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 cossxp 6292 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
31, 2eqsstri 4030 . 2 tpos 𝐹 ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
4 eqid 2737 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
54dmmptss 6261 . . 3 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (dom 𝐹 ∪ {∅})
6 xpss1 5704 . . 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 3993 1 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  cun 3949  wss 3951  c0 4333  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  ccom 5689  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-tpos 8251
This theorem is referenced by:  reltpos  8256  tposexg  8265  wuntpos  10774  catcoppccl  18162
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