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Theorem tposexg 8250
Description: The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
tposexg (𝐹𝑉 → tpos 𝐹 ∈ V)

Proof of Theorem tposexg
StepHypRef Expression
1 tposssxp 8240 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 dmexg 7913 . . . . 5 (𝐹𝑉 → dom 𝐹 ∈ V)
3 cnvexg 7936 . . . . 5 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
42, 3syl 17 . . . 4 (𝐹𝑉dom 𝐹 ∈ V)
5 p0ex 5386 . . . 4 {∅} ∈ V
6 unexg 7755 . . . 4 ((dom 𝐹 ∈ V ∧ {∅} ∈ V) → (dom 𝐹 ∪ {∅}) ∈ V)
74, 5, 6sylancl 584 . . 3 (𝐹𝑉 → (dom 𝐹 ∪ {∅}) ∈ V)
8 rnexg 7914 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
97, 8xpexd 7757 . 2 (𝐹𝑉 → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
10 ssexg 5325 . 2 ((tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V)
111, 9, 10sylancr 585 1 (𝐹𝑉 → tpos 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3471  cun 3945  wss 3947  c0 4324  {csn 4630   × cxp 5678  ccnv 5679  dom cdm 5680  ran crn 5681  tpos ctpos 8235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-tpos 8236
This theorem is referenced by:  tposex  8270  oftpos  22372
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