Theorem tpos0 7618

Description: Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tpos0	⊢ tpos ∅ = ∅

Proof of Theorem tpos0

Step	Hyp	Ref	Expression
1		rel0 5445	. . . 4 ⊢ Rel ∅
2		eqid 2797	. . . . 5 ⊢ ∅ = ∅
3		fn0 6220	. . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
4	2, 3	mpbir 223	. . . 4 ⊢ ∅ Fn ∅
5		tposfn2 7610	. . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅))
6	1, 4, 5	mp2 9	. . 3 ⊢ tpos ∅ Fn ◡∅
7		cnv0 5751	. . . 4 ⊢ ◡∅ = ∅
8	7	fneq2i 6195	. . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅)
9	6, 8	mpbi 222	. 2 ⊢ tpos ∅ Fn ∅
10		fn0 6220	. 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅)
11	9, 10	mpbi 222	1 ⊢ tpos ∅ = ∅

Syntax hints:   = wceq 1653  ∅c0 4113  ^◡ccnv 5309  Rel wrel 5315   Fn wfn 6094  tpos ctpos 7587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-fv 6107  df-tpos 7588

This theorem is referenced by:  oppchomfval  16685  oppgplusfval  18087  opprmulfval  18938