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Theorem unxpwdom3 27233
 Description: Weaker version of unxpwdom 7557 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 4989 . . 3
41, 2, 3syl2anc 643 . 2
5 unxpwdom3.ni . . . . . . 7
65adantr 452 . . . . . 6
7 unxpwdom3.av . . . . . . . 8
87ad2antrr 707 . . . . . . 7
9 oveq2 6089 . . . . . . . . . . . . . 14
109eleq1d 2502 . . . . . . . . . . . . 13
1110notbid 286 . . . . . . . . . . . 12
1211rspcv 3048 . . . . . . . . . . 11
1312adantl 453 . . . . . . . . . 10
14 unxpwdom3.ov . . . . . . . . . . . . . 14
15143expa 1153 . . . . . . . . . . . . 13
16 elun 3488 . . . . . . . . . . . . 13
1715, 16sylib 189 . . . . . . . . . . . 12
1817orcomd 378 . . . . . . . . . . 11
1918ord 367 . . . . . . . . . 10
2013, 19syld 42 . . . . . . . . 9
2120impancom 428 . . . . . . . 8
22 unxpwdom3.lc . . . . . . . . . 10
2322ex 424 . . . . . . . . 9
2423adantr 452 . . . . . . . 8
2521, 24dom2d 7148 . . . . . . 7
268, 25mpd 15 . . . . . 6
276, 26mtand 641 . . . . 5
28 dfrex2 2718 . . . . 5
2927, 28sylibr 204 . . . 4
30 simprr 734 . . . . . . 7
31 unxpwdom3.rc . . . . . . . . . . . . . 14
3231ancom1s 781 . . . . . . . . . . . . 13
3332adantllr 700 . . . . . . . . . . . 12
34333impb 1149 . . . . . . . . . . 11
3534riota5OLD 6576 . . . . . . . . . 10
3635anasss 629 . . . . . . . . 9
3736ancoms 440 . . . . . . . 8
3837eqcomd 2441 . . . . . . 7
39 eqeq2 2445 . . . . . . . . . 10
4039riotabidv 6551 . . . . . . . . 9
4140eqeq2d 2447 . . . . . . . 8
4241rspcev 3052 . . . . . . 7
4330, 38, 42syl2anc 643 . . . . . 6
4443expr 599 . . . . 5
4544reximdva 2818 . . . 4
4629, 45mpd 15 . . 3
47 vex 2959 . . . . . . . . 9
48 vex 2959 . . . . . . . . 9
4947, 48op1std 6357 . . . . . . . 8
5049oveq2d 6097 . . . . . . 7
5147, 48op2ndd 6358 . . . . . . 7
5250, 51eqeq12d 2450 . . . . . 6
5352riotabidv 6551 . . . . 5
5453eqeq2d 2447 . . . 4
5554rexxp 5017 . . 3
5646, 55sylibr 204 . 2
574, 56wdomd 7549 1 *
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  cvv 2956   cun 3318  cop 3817   class class class wbr 4212   cxp 4876  cfv 5454  (class class class)co 6081  c1st 6347  c2nd 6348  crio 6542   cdom 7107   * cwdom 7525 This theorem is referenced by:  isnumbasgrplem2  27246 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-wdom 7527
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