Theorem fihashdom 10265

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.

Step	Hyp	Ref	Expression
1		isfi 6532	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 119	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantr 271	. 2 |- ( ( A e. Fin /\ B e. Fin ) -> E. n e. om A ~~ n )
4		isfi 6532	. . . . 5 |- ( B e. Fin <-> E. m e. om B ~~ m )
5	4	biimpi 119	. . . 4 |- ( B e. Fin -> E. m e. om B ~~ m )
6	5	ad2antlr 474	. . 3 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om B ~~ m )
7		simplrr 504	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> A ~~ n )
8		domen1 6612	. . . . 5 |- ( A ~~ n -> ( A ~<_ m <-> n ~<_ m ) )
9	7, 8	syl 14	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ m <-> n ~<_ m ) )
10		simprr 500	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> B ~~ m )
11		domen2 6613	. . . . 5 |- ( B ~~ m -> ( A ~<_ B <-> A ~<_ m ) )
12	10, 11	syl 14	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( A ~<_ B <-> A ~<_ m ) )
13		0zd 8816	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> 0 e. ZZ )
14		eqid 2089	. . . . . 6 |- frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
15		simplrl 503	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n e. om )
16		simprl 499	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m e. om )
17	13, 14, 15, 16	frec2uzled 9890	. . . . 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 6634	. . . . . 6 |- ( ( n e. om /\ m e. om ) -> ( n ~<_ m <-> n C_ m ) )
19	15, 16, 18	syl2anc 404	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( n ~<_ m <-> n C_ m ) )
20	7	ensymd 6554	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> n ~~ A )
21		hashennn 10242	. . . . . . 7 |- ( ( n e. om /\ n ~~ A ) -> (♯ ` A ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` n ) )
22	15, 20, 21	syl2anc 404	. . . . . 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 ) )
23	10	ensymd 6554	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> m ~~ B )
24		hashennn 10242	. . . . . . 7 |- ( ( m e. om /\ m ~~ B ) -> (♯ ` B ) = (frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 ) ` m ) )
25	16, 23, 24	syl2anc 404	. . . . . 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 ) )
26	22, 25	breq12d 3864	. . . . 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 ) ) )
27	17, 19, 26	3bitr4rd 220	. . . 4 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> n ~<_ m ) )
28	9, 12, 27	3bitr4rd 220	. . 3 |- ( ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ B ~~ m ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
29	6, 28	rexlimddv 2494	. 2 |- ( ( ( A e. Fin /\ B e. Fin ) /\ ( n e. om /\ A ~~ n ) ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )
30	3, 29	rexlimddv 2494	1 |- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   E.wrex 2361   C_ wss 3000   class class class wbr 3851   |-> cmpt 3905   omcom 4418   ` cfv 5028  (class class class)co 5666  freccfrec 6169   ~~ cen 6509   ~<_ cdom 6510   Fincfn 6511   0cc0 7404   1c1 7405   + caddc 7407   <_ cle 7577   ZZcz 8804  ♯chash 10237

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-ihash 10238

This theorem is referenced by:  fihashss  10278  phicl2  11522  phibnd  11525