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Theorem nqereq 9302
Description: The function /Q acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nqereq |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( /Q ` A ) = ( /Q ` B ) ) )
Proof of Theorem nqereq
StepHypRef Expression
1 nqercl 9298 . . . . 5 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
213ad2ant1 1015 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) e. Q. )
3 nqercl 9298 . . . . 5 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
433ad2ant2 1016 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` B ) e. Q. )
5 enqer 9288 . . . . . 6 |- ~Q Er ( N. X. N. )
65a1i 11 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ~Q Er ( N. X. N. ) )
7 nqerrel 9299 . . . . . . 7 |- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )
873ad2ant1 1015 . . . . . 6 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> A ~Q ( /Q ` A ) )
9 simp3 996 . . . . . 6 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> A ~Q B )
106, 8, 9ertr3d 7321 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) ~Q B )
11 nqerrel 9299 . . . . . 6 |- ( B e. ( N. X. N. ) -> B ~Q ( /Q ` B ) )
12113ad2ant2 1016 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> B ~Q ( /Q ` B ) )
136, 10, 12ertrd 7319 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) ~Q ( /Q ` B ) )
14 enqeq 9301 . . . 4 |- ( ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. /\ ( /Q ` A ) ~Q ( /Q ` B ) ) -> ( /Q ` A ) = ( /Q ` B ) )
152, 4, 13, 14syl3anc 1226 . . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) = ( /Q ` B ) )
16153expia 1196 . 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B -> ( /Q ` A ) = ( /Q ` B ) ) )
175a1i 11 . . . 4 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> ~Q Er ( N. X. N. ) )
187adantr 463 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q ( /Q ` A ) )
19 simprr 755 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> ( /Q ` A ) = ( /Q ` B ) )
2018, 19breqtrd 4463 . . . 4 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q ( /Q ` B ) )
2111ad2antrl 725 . . . 4 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> B ~Q ( /Q ` B ) )
2217, 20, 21ertr4d 7322 . . 3 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q B )
2322expr 613 . 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( /Q ` A ) = ( /Q ` B ) -> A ~Q B ) )
2416, 23impbid 191 1 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( /Q ` A ) = ( /Q ` B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   class class class wbr 4439   X. cxp 4986   ` cfv 5570   Er wer 7300   N.cnpi 9211   ~Q ceq 9218   Q.cnq 9219   /Qcerq 9221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ni 9239  df-mi 9241  df-lti 9242  df-enq 9278  df-nq 9279  df-erq 9280  df-1nq 9283
This theorem is referenced by:  adderpq  9323  mulerpq  9324  distrnq  9328  recmulnq  9331  ltexnq  9342
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