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Theorem tposssxp 7401
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 7397 . . 3 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
2 cossxp 5696 . . 3 (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
31, 2eqsstri 3668 . 2 tpos 𝐹 ⊆ (dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) × ran 𝐹)
4 eqid 2651 . . . 4 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
54dmmptss 5669 . . 3 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ⊆ (dom 𝐹 ∪ {∅})
6 xpss1 5161 . . 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 3645 1 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  cun 3605  wss 3607  c0 3948  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  ccom 5147  tpos ctpos 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-tpos 7397
This theorem is referenced by:  reltpos  7402  tposexg  7411  wuntpos  9594  catcoppccl  16805
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