Theorem tposf2 6247

Description: The domain and range 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 5347	. . . . . . 7 |- ( F : A --> B -> F Fn A )
2		dffn4 5426	. . . . . . 7 |- ( F Fn A <-> F : A -onto-> ran F )
3	1, 2	sylib 121	. . . . . 6 |- ( F : A --> B -> F : A -onto-> ran F )
4		tposfo2 6246	. . . . . 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 123	. . . 4 |- ( ( Rel A /\ F : A --> B ) -> tpos F : `' A -onto-> ran F )
7		fof 5420	. . . 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 5356	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	9	adantl 275	. . 3 |- ( ( Rel A /\ F : A --> B ) -> ran F C_ B )
11		fss 5359	. . 3 |- ( (tpos F : `' A --> ran F /\ ran F C_ B ) -> tpos F : `' A --> B )
12	8, 10, 11	syl2anc 409	. 2 |- ( ( Rel A /\ F : A --> B ) -> tpos F : `' A --> B )
13	12	ex 114	1 |- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3121   `'ccnv 4610   ran crn 4612   Rel wrel 4616   Fn wfn 5193   -->wf 5194   -onto->wfo 5196  tpos ctpos 6223

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-tpos 6224

This theorem is referenced by:  tposf  6251