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Theorem unxpwdom3 35659
 Description: Weaker version of unxpwdom 8104 where a function is required only to be cancellative, not an injection. and are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into , each row must hit an element of ; by column injectivity, each row can be identified in at least one way by the element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
Hypotheses
Ref Expression
unxpwdom3.av
unxpwdom3.bv
unxpwdom3.dv
unxpwdom3.ov
unxpwdom3.lc
unxpwdom3.rc
unxpwdom3.ni
Assertion
Ref Expression
unxpwdom3 *
Distinct variable groups:   ,,,,   ,,,,   ,,,,   ,,,,   ,,,,   ,,
Allowed substitution hints:   (,)   (,,,)   (,,,)   (,,,)

Proof of Theorem unxpwdom3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unxpwdom3.dv . . 3
2 unxpwdom3.bv . . 3
3 xpexg 6607 . . 3
41, 2, 3syl2anc 665 . 2
5 simprr 764 . . . . 5
6 simplr 760 . . . . . . 7
7 unxpwdom3.rc . . . . . . . . . 10
87an4s 833 . . . . . . . . 9
98anassrs 652 . . . . . . . 8
109adantlrr 725 . . . . . . 7
116, 10riota5 6292 . . . . . 6
1211eqcomd 2437 . . . . 5
13 eqeq2 2444 . . . . . . . 8
1413riotabidv 6269 . . . . . . 7
1514eqeq2d 2443 . . . . . 6
1615rspcev 3188 . . . . 5
175, 12, 16syl2anc 665 . . . 4
18 unxpwdom3.ni . . . . . . 7
1918adantr 466 . . . . . 6
20 unxpwdom3.av . . . . . . . 8
2120ad2antrr 730 . . . . . . 7
22 oveq2 6313 . . . . . . . . . . . . . 14
2322eleq1d 2498 . . . . . . . . . . . . 13
2423notbid 295 . . . . . . . . . . . 12
2524rspcv 3184 . . . . . . . . . . 11
2625adantl 467 . . . . . . . . . 10
27 unxpwdom3.ov . . . . . . . . . . . . . 14
28273expa 1205 . . . . . . . . . . . . 13
29 elun 3612 . . . . . . . . . . . . 13
3028, 29sylib 199 . . . . . . . . . . . 12
3130orcomd 389 . . . . . . . . . . 11
3231ord 378 . . . . . . . . . 10
3326, 32syld 45 . . . . . . . . 9
3433impancom 441 . . . . . . . 8
35 unxpwdom3.lc . . . . . . . . . 10
3635ex 435 . . . . . . . . 9
3736adantr 466 . . . . . . . 8
3834, 37dom2d 7617 . . . . . . 7
3921, 38mpd 15 . . . . . 6
4019, 39mtand 663 . . . . 5
41 dfrex2 2883 . . . . 5
4240, 41sylibr 215 . . . 4
4317, 42reximddv 2908 . . 3
44 vex 3090 . . . . . . . . 9
45 vex 3090 . . . . . . . . 9
4644, 45op1std 6817 . . . . . . . 8
4746oveq2d 6321 . . . . . . 7
4844, 45op2ndd 6818 . . . . . . 7
4947, 48eqeq12d 2451 . . . . . 6
5049riotabidv 6269 . . . . 5
5150eqeq2d 2443 . . . 4
5251rexxp 4997 . . 3
5343, 52sylibr 215 . 2
544, 53wdomd 8096 1 *
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   w3a 982   wceq 1437   wcel 1870  wral 2782  wrex 2783  cvv 3087   cun 3440  cop 4008   class class class wbr 4426   cxp 4852  cfv 5601  crio 6266  (class class class)co 6305  c1st 6805  c2nd 6806   cdom 7575   * cwdom 8072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-1st 6807  df-2nd 6808  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-wdom 8074 This theorem is referenced by:  isnumbasgrplem2  35669
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