Theorem tposfn2 7374

Description: The domain of a transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposfn2	|- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )

Proof of Theorem tposfn2

Step	Hyp	Ref	Expression
1		tposfun 7368	. . . 4 |- ( Fun F -> Fun tpos F )
2	1	a1i 11	. . 3 |- ( Rel A -> ( Fun F -> Fun tpos F ) )
3		dmtpos 7364	. . . . . 6 |- ( Rel dom F -> dom tpos F = `' dom F )
4	3	a1i 11	. . . . 5 |- ( dom F = A -> ( Rel dom F -> dom tpos F = `' dom F ) )
5		releq 5201	. . . . 5 |- ( dom F = A -> ( Rel dom F <-> Rel A ) )
6		cnveq 5296	. . . . . 6 |- ( dom F = A -> `' dom F = `' A )
7	6	eqeq2d 2632	. . . . 5 |- ( dom F = A -> ( dom tpos F = `' dom F <-> dom tpos F = `' A ) )
8	4, 5, 7	3imtr3d 282	. . . 4 |- ( dom F = A -> ( Rel A -> dom tpos F = `' A ) )
9	8	com12 32	. . 3 |- ( Rel A -> ( dom F = A -> dom tpos F = `' A ) )
10	2, 9	anim12d 586	. 2 |- ( Rel A -> ( ( Fun F /\ dom F = A ) -> ( Fun tpos F /\ dom tpos F = `' A ) ) )
11		df-fn 5891	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
12		df-fn 5891	. 2 |- (tpos F Fn `' A <-> ( Fun tpos F /\ dom tpos F = `' A ) )
13	10, 11, 12	3imtr4g 285	1 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   `'ccnv 5113   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   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:  tposfo2  7375  tpos0  7382