Theorem tposfn2 6171

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 6165	. . . 4 |- ( Fun F -> Fun tpos F )
2	1	a1i 9	. . 3 |- ( Rel A -> ( Fun F -> Fun tpos F ) )
3		dmtpos 6161	. . . . . 6 |- ( Rel dom F -> dom tpos F = `' dom F )
4	3	a1i 9	. . . . 5 |- ( dom F = A -> ( Rel dom F -> dom tpos F = `' dom F ) )
5		releq 4629	. . . . 5 |- ( dom F = A -> ( Rel dom F <-> Rel A ) )
6		cnveq 4721	. . . . . 6 |- ( dom F = A -> `' dom F = `' A )
7	6	eqeq2d 2152	. . . . 5 |- ( dom F = A -> ( dom tpos F = `' dom F <-> dom tpos F = `' A ) )
8	4, 5, 7	3imtr3d 201	. . . 4 |- ( dom F = A -> ( Rel A -> dom tpos F = `' A ) )
9	8	com12 30	. . 3 |- ( Rel A -> ( dom F = A -> dom tpos F = `' A ) )
10	2, 9	anim12d 333	. 2 |- ( Rel A -> ( ( Fun F /\ dom F = A ) -> ( Fun tpos F /\ dom tpos F = `' A ) ) )
11		df-fn 5134	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
12		df-fn 5134	. 2 |- (tpos F Fn `' A <-> ( Fun tpos F /\ dom tpos F = `' A ) )
13	10, 11, 12	3imtr4g 204	1 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   `'ccnv 4546   dom cdm 4547   Rel wrel 4552   Fun wfun 5125   Fn wfn 5126  tpos ctpos 6149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-tpos 6150

This theorem is referenced by:  tposfo2  6172  tpos0  6179