Theorem tposfn2 6333

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 6327	. . . 4 |- ( Fun F -> Fun tpos F )
2	1	a1i 9	. . 3 |- ( Rel A -> ( Fun F -> Fun tpos F ) )
3		dmtpos 6323	. . . . . 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 4746	. . . . 5 |- ( dom F = A -> ( Rel dom F <-> Rel A ) )
6		cnveq 4841	. . . . . 6 |- ( dom F = A -> `' dom F = `' A )
7	6	eqeq2d 2208	. . . . 5 |- ( dom F = A -> ( dom tpos F = `' dom F <-> dom tpos F = `' A ) )
8	4, 5, 7	3imtr3d 202	. . . 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 335	. 2 |- ( Rel A -> ( ( Fun F /\ dom F = A ) -> ( Fun tpos F /\ dom tpos F = `' A ) ) )
11		df-fn 5262	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
12		df-fn 5262	. 2 |- (tpos F Fn `' A <-> ( Fun tpos F /\ dom tpos F = `' A ) )
13	10, 11, 12	3imtr4g 205	1 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   `'ccnv 4663   dom cdm 4664   Rel wrel 4669   Fun wfun 5253   Fn wfn 5254  tpos ctpos 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-tpos 6312

This theorem is referenced by:  tposfo2  6334  tpos0  6341