Theorem tposf2 6377

Description: The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)

Assertion

Ref	Expression
tposf2	|- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )

Proof of Theorem tposf2

Step	Hyp	Ref	Expression
1		ffn 5445	. . . . . . 7 |- ( F : A --> B -> F Fn A )
2		dffn4 5526	. . . . . . 7 |- ( F Fn A <-> F : A -onto-> ran F )
3	1, 2	sylib 122	. . . . . 6 |- ( F : A --> B -> F : A -onto-> ran F )
4		tposfo2 6376	. . . . . 6 |- ( Rel A -> ( F : A -onto-> ran F -> tpos F : `' A -onto-> ran F ) )
5	3, 4	syl5 32	. . . . 5 |- ( Rel A -> ( F : A --> B -> tpos F : `' A -onto-> ran F ) )
6	5	imp 124	. . . 4 |- ( ( Rel A /\ F : A --> B ) -> tpos F : `' A -onto-> ran F )
7		fof 5520	. . . 4 |- (tpos F : `' A -onto-> ran F -> tpos F : `' A --> ran F )
8	6, 7	syl 14	. . 3 |- ( ( Rel A /\ F : A --> B ) -> tpos F : `' A --> ran F )
9		frn 5454	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	9	adantl 277	. . 3 |- ( ( Rel A /\ F : A --> B ) -> ran F C_ B )
11		fss 5457	. . 3 |- ( (tpos F : `' A --> ran F /\ ran F C_ B ) -> tpos F : `' A --> B )
12	8, 10, 11	syl2anc 411	. 2 |- ( ( Rel A /\ F : A --> B ) -> tpos F : `' A --> B )
13	12	ex 115	1 |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3174   `'ccnv 4692   ran crn 4694   Rel wrel 4698   Fn wfn 5285   -->wf 5286   -onto->wfo 5288  tpos ctpos 6353

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-tpos 6354

This theorem is referenced by:  tposf  6381