Theorem tposfn2 6475

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 6469	. . . 4 |- ( Fun F -> Fun tpos F )
2	1	a1i 9	. . 3 |- ( Rel A -> ( Fun F -> Fun tpos F ) )
3		dmtpos 6465	. . . . . 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 4814	. . . . 5 |- ( dom F = A -> ( Rel dom F <-> Rel A ) )
6		cnveq 4910	. . . . . 6 |- ( dom F = A -> `' dom F = `' A )
7	6	eqeq2d 2243	. . . . 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 5336	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
12		df-fn 5336	. 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 1398   `'ccnv 4730   dom cdm 4731   Rel wrel 4736   Fun wfun 5327   Fn wfn 5328  tpos ctpos 6453

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-tpos 6454

This theorem is referenced by:  tposfo2  6476  tpos0  6483