Theorem tpos0 7382

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 5243	. . . 4 |- Rel (/)
2		eqid 2622	. . . . 5 |- (/) = (/)
3		fn0 6011	. . . . 5 |- ( (/) Fn (/) <-> (/) = (/) )
4	2, 3	mpbir 221	. . . 4 |- (/) Fn (/)
5		tposfn2 7374	. . . 4 |- ( Rel (/) -> ( (/) Fn (/) -> tpos (/) Fn `' (/) ) )
6	1, 4, 5	mp2 9	. . 3 |- tpos (/) Fn `' (/)
7		cnv0 5535	. . . 4 |- `' (/) = (/)
8	7	fneq2i 5986	. . 3 |- (tpos (/) Fn `' (/) <-> tpos (/) Fn (/) )
9	6, 8	mpbi 220	. 2 |- tpos (/) Fn (/)
10		fn0 6011	. 2 |- (tpos (/) Fn (/) <-> tpos (/) = (/) )
11	9, 10	mpbi 220	1 |- tpos (/) = (/)

Syntax hints:   = wceq 1483   (/)c0 3915   `'ccnv 5113   Rel wrel 5119   Fn wfn 5883  tpos ctpos 7351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-tpos 7352

This theorem is referenced by:  oppchomfval  16374  oppgplusfval  17778  opprmulfval  18625