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Theorem unxpwdom3 26422
 Description: Weaker version of unxpwdom 7187 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
StepHypRef Expression
1 unxpwdom3.dv . . 3
2 unxpwdom3.bv . . 3
3 xpexg 4707 . . 3
41, 2, 3syl2anc 645 . 2
5 unxpwdom3.ni . . . . . . 7
65adantr 453 . . . . . 6
7 unxpwdom3.av . . . . . . . 8
87ad2antrr 709 . . . . . . 7
9 oveq2 5718 . . . . . . . . . . . . . 14
109eleq1d 2319 . . . . . . . . . . . . 13
1110notbid 287 . . . . . . . . . . . 12
1211rcla4v 2817 . . . . . . . . . . 11
1312adantl 454 . . . . . . . . . 10
14 unxpwdom3.ov . . . . . . . . . . . . . 14
15143expa 1156 . . . . . . . . . . . . 13
16 elun 3226 . . . . . . . . . . . . 13
1715, 16sylib 190 . . . . . . . . . . . 12
1817orcomd 379 . . . . . . . . . . 11
1918ord 368 . . . . . . . . . 10
2013, 19syld 42 . . . . . . . . 9
2120impancom 429 . . . . . . . 8
22 unxpwdom3.lc . . . . . . . . . 10
2322ex 425 . . . . . . . . 9
2423adantr 453 . . . . . . . 8
2521, 24dom2d 6788 . . . . . . 7
268, 25mpd 16 . . . . . 6
276, 26mtand 643 . . . . 5
28 dfrex2 2520 . . . . 5
2927, 28sylibr 205 . . . 4
30 simprr 736 . . . . . . 7
31 unxpwdom3.rc . . . . . . . . . . . . . 14
3231ancom1s 783 . . . . . . . . . . . . 13
3332adantllr 702 . . . . . . . . . . . 12
34333impb 1152 . . . . . . . . . . 11
3534riota5OLD 6217 . . . . . . . . . 10
3635anasss 631 . . . . . . . . 9
3736ancoms 441 . . . . . . . 8
3837eqcomd 2258 . . . . . . 7
39 eqeq2 2262 . . . . . . . . . 10
4039riotabidv 6192 . . . . . . . . 9
4140eqeq2d 2264 . . . . . . . 8
4241rcla4ev 2821 . . . . . . 7
4330, 38, 42syl2anc 645 . . . . . 6
4443expr 601 . . . . 5
4544reximdva 2617 . . . 4
4629, 45mpd 16 . . 3
47 vex 2730 . . . . . . . . 9
48 vex 2730 . . . . . . . . 9
4947, 48op1std 5982 . . . . . . . 8
5049oveq2d 5726 . . . . . . 7
5147, 48op2ndd 5983 . . . . . . 7
5250, 51eqeq12d 2267 . . . . . 6
5352riotabidv 6192 . . . . 5
5453eqeq2d 2264 . . . 4
5554rexxp 4735 . . 3
5646, 55sylibr 205 . 2
574, 56wdomd 7179 1 *
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wo 359   wa 360   w3a 939   wceq 1619   wcel 1621  wral 2509  wrex 2510  cvv 2727   cun 3076  cop 3547   class class class wbr 3920   cxp 4578  cfv 4592  (class class class)co 5710  c1st 5972  c2nd 5973  crio 6181   cdom 6747   * cwdom 7155 This theorem is referenced by:  isnumbasgrplem2  26435 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-wdom 7157
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