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Theorem tposexg 6252
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 6243 . 2 tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹)
2 dmexg 4886 . . . . 5 (𝐹𝑉 → dom 𝐹 ∈ V)
3 cnvexg 5161 . . . . 5 (dom 𝐹 ∈ V → dom 𝐹 ∈ V)
42, 3syl 14 . . . 4 (𝐹𝑉dom 𝐹 ∈ V)
5 p0ex 4185 . . . 4 {∅} ∈ V
6 unexg 4439 . . . 4 ((dom 𝐹 ∈ V ∧ {∅} ∈ V) → (dom 𝐹 ∪ {∅}) ∈ V)
74, 5, 6sylancl 413 . . 3 (𝐹𝑉 → (dom 𝐹 ∪ {∅}) ∈ V)
8 rnexg 4887 . . 3 (𝐹𝑉 → ran 𝐹 ∈ V)
9 xpexg 4736 . . 3 (((dom 𝐹 ∪ {∅}) ∈ V ∧ ran 𝐹 ∈ V) → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
107, 8, 9syl2anc 411 . 2 (𝐹𝑉 → ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V)
11 ssexg 4139 . 2 ((tpos 𝐹 ⊆ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∧ ((dom 𝐹 ∪ {∅}) × ran 𝐹) ∈ V) → tpos 𝐹 ∈ V)
121, 10, 11sylancr 414 1 (𝐹𝑉 → tpos 𝐹 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737  cun 3127  wss 3129  c0 3422  {csn 3591   × cxp 4620  ccnv 4621  dom cdm 4622  ran crn 4623  tpos ctpos 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-tpos 6239
This theorem is referenced by:  tposex  6272  opprvalg  13053  opprmulfvalg  13054  opprex  13057  opprsllem  13058
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