Theorem tposfo2 6235

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 6234	. . . 4 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
2	1	adantrd 277	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> tpos F Fn `' A ) )
3		fndm 5287	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
4	3	releqd 4688	. . . . . . . 8 |- ( F Fn A -> ( Rel dom F <-> Rel A ) )
5	4	biimparc 297	. . . . . . 7 |- ( ( Rel A /\ F Fn A ) -> Rel dom F )
6		rntpos 6225	. . . . . . 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 2174	. . . . 5 |- ( ( Rel A /\ F Fn A ) -> ( ran tpos F = B <-> ran F = B ) )
9	8	biimprd 157	. . . 4 |- ( ( Rel A /\ F Fn A ) -> ( ran F = B -> ran tpos F = B ) )
10	9	expimpd 361	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> ran tpos F = B ) )
11	2, 10	jcad 305	. 2 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> (tpos F Fn `' A /\ ran tpos F = B ) ) )
12		df-fo 5194	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
13		df-fo 5194	. 2 |- (tpos F : `' A -onto-> B <-> (tpos F Fn `' A /\ ran tpos F = B ) )
14	11, 12, 13	3imtr4g 204	1 |- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   `'ccnv 4603   dom cdm 4604   ran crn 4605   Rel wrel 4609   Fn wfn 5183   -onto->wfo 5186  tpos ctpos 6212

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fo 5194  df-fv 5196  df-tpos 6213

This theorem is referenced by:  tposf2  6236  tposf1o2  6238  tposfo  6239