Theorem tposfo2 6322

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 6321	. . . 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 5354	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
4	3	releqd 4744	. . . . . . . 8 |- ( F Fn A -> ( Rel dom F <-> Rel A ) )
5	4	biimparc 299	. . . . . . 7 |- ( ( Rel A /\ F Fn A ) -> Rel dom F )
6		rntpos 6312	. . . . . . 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 2202	. . . . 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 5261	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
13		df-fo 5261	. 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 1364   `'ccnv 4659   dom cdm 4660   ran crn 4661   Rel wrel 4665   Fn wfn 5250   -onto->wfo 5253  tpos ctpos 6299

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fo 5261  df-fv 5263  df-tpos 6300

This theorem is referenced by:  tposf2  6323  tposf1o2  6325  tposfo  6326