Theorem tposfn2 6163

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 6157	. . . 4 |- ( Fun F -> Fun tpos F )
2	1	a1i 9	. . 3 |- ( Rel A -> ( Fun F -> Fun tpos F ) )
3		dmtpos 6153	. . . . . 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 4621	. . . . 5 |- ( dom F = A -> ( Rel dom F <-> Rel A ) )
6		cnveq 4713	. . . . . 6 |- ( dom F = A -> `' dom F = `' A )
7	6	eqeq2d 2151	. . . . 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 5126	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
12		df-fn 5126	. 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 1331   `'ccnv 4538   dom cdm 4539   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118  tpos ctpos 6141

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-tpos 6142

This theorem is referenced by:  tposfo2  6164  tpos0  6171