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Theorem nqereq 9757
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 9753 . . . . 5 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
213ad2ant1 1082 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) e. Q. )
3 nqercl 9753 . . . . 5 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
433ad2ant2 1083 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` B ) e. Q. )
5 enqer 9743 . . . . . 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 9754 . . . . . . 7 |- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )
873ad2ant1 1082 . . . . . 6 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> A ~Q ( /Q ` A ) )
9 simp3 1063 . . . . . 6 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> A ~Q B )
106, 8, 9ertr3d 7760 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) ~Q B )
11 nqerrel 9754 . . . . . 6 |- ( B e. ( N. X. N. ) -> B ~Q ( /Q ` B ) )
12113ad2ant2 1083 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> B ~Q ( /Q ` B ) )
136, 10, 12ertrd 7758 . . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) ~Q ( /Q ` B ) )
14 enqeq 9756 . . . 4 |- ( ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. /\ ( /Q ` A ) ~Q ( /Q ` B ) ) -> ( /Q ` A ) = ( /Q ` B ) )
152, 4, 13, 14syl3anc 1326 . . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ A ~Q B ) -> ( /Q ` A ) = ( /Q ` B ) )
16153expia 1267 . 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 481 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q ( /Q ` A ) )
19 simprr 796 . . . . 5 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> ( /Q ` A ) = ( /Q ` B ) )
2018, 19breqtrd 4679 . . . 4 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q ( /Q ` B ) )
2111ad2antrl 764 . . . 4 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> B ~Q ( /Q ` B ) )
2217, 20, 21ertr4d 7761 . . 3 |- ( ( A e. ( N. X. N. ) /\ ( B e. ( N. X. N. ) /\ ( /Q ` A ) = ( /Q ` B ) ) ) -> A ~Q B )
2322expr 643 . 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( /Q ` A ) = ( /Q ` B ) -> A ~Q B ) )
2416, 23impbid 202 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   X. cxp 5112   ` cfv 5888   Er wer 7739   N.cnpi 9666   ~Q ceq 9673   Q.cnq 9674   /Qcerq 9676
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-3or 1038  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-enq 9733  df-nq 9734  df-erq 9735  df-1nq 9738
This theorem is referenced by:  adderpq  9778  mulerpq  9779  distrnq  9783  recmulnq  9786  ltexnq  9797
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