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Theorem fihashdom 11025
Description: Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
Assertion
Ref Expression
fihashdom |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
Proof of Theorem fihashdom
Dummy variables m n x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6912 . . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
21biimpi 120 . . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
32adantr 276 . 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. n e. om A ~~ n )
4 isfi 6912 . . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
54biimpi 120 . . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
65ad2antlr 489 . . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7 simplrr 536 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
8 domen1 7003 . . . . 5 |- ( A ~~ n -> ( A ~<_ m <-> n ~<_ m ) )
97, 8syl 14 . . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ m <-> n ~<_ m ) )
10 simprr 531 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
11 domen2 7004 . . . . 5 |- ( B ~~ m -> ( A ~<_ B <-> A ~<_ m ) )
1210, 11syl 14 . . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B <-> A ~<_ m ) )
13 0zd 9458 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
14 eqid 2229 . . . . . 6 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
15 simplrl 535 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
16 simprl 529 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
1713, 14, 15, 16frec2uzled 10651 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n C_ m <-> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) <_ (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) ) )
18 nndomo 7025 . . . . . 6 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
1915, 16, 18syl2anc 411 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n ~<_ m <-> n C_ m ) )
207ensymd 6935 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
21 hashennn 11002 . . . . . . 7 |- ( ( n e. om /\ n ~~ A ) -> (♯ ` A ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
2215, 20, 21syl2anc 411 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> (♯ ` A ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
2310ensymd 6935 . . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
24 hashennn 11002 . . . . . . 7 |- ( ( m e. om /\ m ~~ B ) -> (♯ ` B ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
2516, 23, 24syl2anc 411 . . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> (♯ ` B ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
2622, 25breq12d 4096 . . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) <_ (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) ) )
2717, 19, 263bitr4rd 221 . . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> n ~<_ m ) )
289, 12, 273bitr4rd 221 . . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
296, 28rexlimddv 2653 . 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
303, 29rexlimddv 2653 1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   class class class wbr 4083   |-> cmpt 4145   omcom 4682   ` cfv 5318  (class class class)co 6001  freccfrec 6536   ~~ cen 6885   ~<_ cdom 6886   Fincfn 6887   0cc0 7999   1c1 8000   + caddc 8002   <_ cle 8182   ZZcz 9446  ♯chash 10997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-recs 6451  df-frec 6537  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-ihash 10998
This theorem is referenced by:  fihashss  11038  phicl2  12736  phibnd  12739  4sqlem11  12924  znidom  14621  znidomb  14622
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