Theorem tposfo2 6270

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 6269	. . . 4 |- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
2	1	adantrd 279	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> tpos F Fn `' A ) )
3		fndm 5317	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
4	3	releqd 4712	. . . . . . . 8 |- ( F Fn A -> ( Rel dom F <-> Rel A ) )
5	4	biimparc 299	. . . . . . 7 |- ( ( Rel A /\ F Fn A ) -> Rel dom F )
6		rntpos 6260	. . . . . . 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 2186	. . . . 5 |- ( ( Rel A /\ F Fn A ) -> ( ran tpos F = B <-> ran F = B ) )
9	8	biimprd 158	. . . 4 |- ( ( Rel A /\ F Fn A ) -> ( ran F = B -> ran tpos F = B ) )
10	9	expimpd 363	. . 3 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> ran tpos F = B ) )
11	2, 10	jcad 307	. 2 |- ( Rel A -> ( ( F Fn A /\ ran F = B ) -> (tpos F Fn `' A /\ ran tpos F = B ) ) )
12		df-fo 5224	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
13		df-fo 5224	. 2 |- (tpos F : `' A -onto-> B <-> (tpos F Fn `' A /\ ran tpos F = B ) )
14	11, 12, 13	3imtr4g 205	1 |- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   `'ccnv 4627   dom cdm 4628   ran crn 4629   Rel wrel 4633   Fn wfn 5213   -onto->wfo 5216  tpos ctpos 6247

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fo 5224  df-fv 5226  df-tpos 6248

This theorem is referenced by:  tposf2  6271  tposf1o2  6273  tposfo  6274