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Theorem tposexg 7764
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 7754 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 dmexg 7476 . . . . 5 (𝐹𝑉 → dom 𝐹 ∈ V)
3 cnvexg 7492 . . . . 5 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
42, 3syl 17 . . . 4 (𝐹𝑉dom 𝐹 ∈ V)
5 p0ex 5182 . . . 4 {∅} ∈ V
6 unexg 7336 . . . 4 ((dom 𝐹 ∈ V ∧ {∅} ∈ V) → (dom 𝐹 ∪ {∅}) ∈ V)
74, 5, 6sylancl 586 . . 3 (𝐹𝑉 → (dom 𝐹 ∪ {∅}) ∈ V)
8 rnexg 7477 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
97, 8xpexd 7338 . 2 (𝐹𝑉 → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
10 ssexg 5125 . 2 ((tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V)
111, 9, 10sylancr 587 1 (𝐹𝑉 → tpos 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2083  Vcvv 3440  cun 3863  wss 3865  c0 4217  {csn 4478   × cxp 5448  ccnv 5449  dom cdm 5450  ran crn 5451  tpos ctpos 7749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-tpos 7750
This theorem is referenced by:  tposex  7784  oftpos  20749
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