Theorem tposfo2 6428

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 6427	. . . 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 5426	. . . . . . . . 9 |- ( F Fn A -> dom F = A )
4	3	releqd 4808	. . . . . . . 8 |- ( F Fn A -> ( Rel dom F <-> Rel A ) )
5	4	biimparc 299	. . . . . . 7 |- ( ( Rel A /\ F Fn A ) -> Rel dom F )
6		rntpos 6418	. . . . . . 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 2238	. . . . 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 5330	. 2 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
13		df-fo 5330	. 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 1395   `'ccnv 4722   dom cdm 4723   ran crn 4724   Rel wrel 4728   Fn wfn 5319   -onto->wfo 5322  tpos ctpos 6405

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fo 5330  df-fv 5332  df-tpos 6406

This theorem is referenced by:  tposf2  6429  tposf1o2  6431  tposfo  6432