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Mirrors > Home > MPE Home > Th. List > tposfo2 | Structured version Visualization version Unicode version |
Description: Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposfo2 | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposfn2 7374 | . . . 4 tpos | |
2 | 1 | adantrd 484 | . . 3 tpos |
3 | fndm 5990 | . . . . . . . . 9 | |
4 | 3 | releqd 5203 | . . . . . . . 8 |
5 | 4 | biimparc 504 | . . . . . . 7 |
6 | rntpos 7365 | . . . . . . 7 tpos | |
7 | 5, 6 | syl 17 | . . . . . 6 tpos |
8 | 7 | eqeq1d 2624 | . . . . 5 tpos |
9 | 8 | biimprd 238 | . . . 4 tpos |
10 | 9 | expimpd 629 | . . 3 tpos |
11 | 2, 10 | jcad 555 | . 2 tpos tpos |
12 | df-fo 5894 | . 2 | |
13 | df-fo 5894 | . 2 tpos tpos tpos | |
14 | 11, 12, 13 | 3imtr4g 285 | 1 tpos |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 ccnv 5113 cdm 5114 crn 5115 wrel 5119 wfn 5883 wfo 5886 tpos ctpos 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fo 5894 df-fv 5896 df-tpos 7352 |
This theorem is referenced by: tposf2 7376 tposf1o2 7378 tposfo 7379 oppglsm 18057 |
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