Theorem tposf2 6259

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 5357	. . . . . . 7 |- ( F : A --> B -> F Fn A )
2		dffn4 5436	. . . . . . 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 6258	. . . . . 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 5430	. . . 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 5366	. . . 4 |- ( F : A --> B -> ran F C_ B )
10	9	adantl 277	. . 3 |- ( ( Rel A /\ F : A --> B ) -> ran F C_ B )
11		fss 5369	. . 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 3127   `'ccnv 4619   ran crn 4621   Rel wrel 4625   Fn wfn 5203   -->wf 5204   -onto->wfo 5206  tpos ctpos 6235

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fo 5214  df-fv 5216  df-tpos 6236

This theorem is referenced by:  tposf  6263