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Theorem tposexg 8175
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 8165 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 dmexg 7844 . . . . 5 (𝐹𝑉 → dom 𝐹 ∈ V)
3 cnvexg 7865 . . . . 5 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
42, 3syl 17 . . . 4 (𝐹𝑉dom 𝐹 ∈ V)
5 p0ex 5343 . . . 4 {∅} ∈ V
6 unexg 7687 . . . 4 ((dom 𝐹 ∈ V ∧ {∅} ∈ V) → (dom 𝐹 ∪ {∅}) ∈ V)
74, 5, 6sylancl 587 . . 3 (𝐹𝑉 → (dom 𝐹 ∪ {∅}) ∈ V)
8 rnexg 7845 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
97, 8xpexd 7689 . 2 (𝐹𝑉 → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
10 ssexg 5284 . 2 ((tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V)
111, 9, 10sylancr 588 1 (𝐹𝑉 → tpos 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3447  cun 3912  wss 3914  c0 4286  {csn 4590   × cxp 5635  ccnv 5636  dom cdm 5637  ran crn 5638  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-tpos 8161
This theorem is referenced by:  tposex  8195  oftpos  21824
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