Theorem tpos0 7533

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 5382	. . . 4 ⊢ Rel ∅
2		eqid 2770	. . . . 5 ⊢ ∅ = ∅
3		fn0 6151	. . . . 5 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
4	2, 3	mpbir 221	. . . 4 ⊢ ∅ Fn ∅
5		tposfn2 7525	. . . 4 ⊢ (Rel ∅ → (∅ Fn ∅ → tpos ∅ Fn ◡∅))
6	1, 4, 5	mp2 9	. . 3 ⊢ tpos ∅ Fn ◡∅
7		cnv0 5676	. . . 4 ⊢ ◡∅ = ∅
8	7	fneq2i 6126	. . 3 ⊢ (tpos ∅ Fn ◡∅ ↔ tpos ∅ Fn ∅)
9	6, 8	mpbi 220	. 2 ⊢ tpos ∅ Fn ∅
10		fn0 6151	. 2 ⊢ (tpos ∅ Fn ∅ ↔ tpos ∅ = ∅)
11	9, 10	mpbi 220	1 ⊢ tpos ∅ = ∅

Syntax hints:   = wceq 1630  ∅c0 4061  ^◡ccnv 5248  Rel wrel 5254   Fn wfn 6026  tpos ctpos 7502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-tpos 7503

This theorem is referenced by:  oppchomfval  16580  oppgplusfval  17984  opprmulfval  18832