Theorem tposfo2 5882

Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposfo2	|- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )

Proof of Theorem tposfo2

Step	Hyp	Ref	Expression
1		tposfn2 5881	. . . 4 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
2	1	adantrd 264	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> tpos F Fn `' A ) )
3		fndm 4998	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
4	3	releqd 4424	. . . . . . . 8 |- ( F Fn A -> ( Rel dom F <-> Rel A ) )
5	4	biimparc 283	. . . . . . 7 |- ( ( Rel A /\ F Fn A ) -> Rel dom F )
6		rntpos 5872	. . . . . . 7 |- ( Rel dom F -> ran tpos F = ran F )
7	5, 6	syl 14	. . . . . 6 |- ( ( Rel A /\ F Fn A ) -> ran tpos F = ran F )
8	7	eqeq1d 2048	. . . . 5 |- ( ( Rel A /\ F Fn A ) -> ( ran tpos F = B <-> ran F = B ) )
9	8	biimprd 147	. . . 4 |- ( ( Rel A /\ F Fn A ) -> ( ran F = B -> ran tpos F = B ) )
10	9	expimpd 345	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> ran tpos F = B ) )
11	2, 10	jcad 291	. 2 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> (tpos F Fn `' A /\ ran tpos F = B ) ) )
12		df-fo 4908	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
13		df-fo 4908	. 2 |- (tpos F : `' A -onto-> B <-> (tpos F Fn `' A /\ ran tpos F = B ) )
14	11, 12, 13	3imtr4g 194	1 |- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   `'ccnv 4344   dom cdm 4345   ran crn 4346   Rel wrel 4350   Fn wfn 4897   -onto->wfo 4900  tpos ctpos 5859

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fo 4908  df-fv 4910  df-tpos 5860

This theorem is referenced by:  tposf2  5883  tposf1o2  5885  tposfo  5886