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Theorem tposexg 7318
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 7308 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 dmexg 7051 . . . . 5 (𝐹𝑉 → dom 𝐹 ∈ V)
3 cnvexg 7066 . . . . 5 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
42, 3syl 17 . . . 4 (𝐹𝑉dom 𝐹 ∈ V)
5 p0ex 4818 . . . 4 {∅} ∈ V
6 unexg 6919 . . . 4 ((dom 𝐹 ∈ V ∧ {∅} ∈ V) → (dom 𝐹 ∪ {∅}) ∈ V)
74, 5, 6sylancl 693 . . 3 (𝐹𝑉 → (dom 𝐹 ∪ {∅}) ∈ V)
8 rnexg 7052 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
9 xpexg 6920 . . 3 (((dom 𝐹 ∪ {∅}) ∈ V ∧ ran 𝐹 ∈ V) → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
107, 8, 9syl2anc 692 . 2 (𝐹𝑉 → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
11 ssexg 4769 . 2 ((tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V)
121, 10, 11sylancr 694 1 (𝐹𝑉 → tpos 𝐹 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3189  cun 3557  wss 3559  c0 3896  {csn 4153   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  tpos ctpos 7303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-tpos 7304
This theorem is referenced by:  tposex  7338  oftpos  20190
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