Theorem ennnfonelemf1 11776

Description: Lemma for ennnfone 11783. L is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemh.dceq	|- ( ph -> A. x e. A A. y e. A DECID x = y )
ennnfonelemh.f	|- ( ph -> F : om -onto-> A )
ennnfonelemh.ne	|- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
ennnfonelemh.g	|- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )
ennnfonelemh.n	|- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
ennnfonelemh.j	|- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )
ennnfonelemh.h	|- H = seq 0 ( G , J )
ennnfone.l	|- L = U_ i e. NN0 ( H ` i )

Assertion

Ref	Expression
ennnfonelemf1	|- ( ph -> L : dom L -1-1-> A )

Distinct variable groups:   A, j, x, y   x, F, y, j, k   n, F   j, G   i, H   j, H, x, y, k   j, J   x, N, y, k, j   ph, j, x, y, k   k, n, j

Allowed substitution hints:   ph( i, n)   A( i, k, n)   F( i)   G( x, y, i, k, n)   H( n)   J( x, y, i, k, n)   L( x, y, i, j, k, n)   N( i, n)

Proof of Theorem ennnfonelemf1

Dummy variables q s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . . 5 |- ( ph -> A. x e. A A. y e. A DECID x = y )
2		ennnfonelemh.f	. . . . 5 |- ( ph -> F : om -onto-> A )
3		ennnfonelemh.ne	. . . . 5 |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
4		ennnfonelemh.g	. . . . 5 |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )
5		ennnfonelemh.n	. . . . 5 |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
6		ennnfonelemh.j	. . . . 5 |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )
7		ennnfonelemh.h	. . . . 5 |- H = seq 0 ( G , J )
8		ennnfone.l	. . . . 5 |- L = U_ i e. NN0 ( H ` i )
9	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemfun 11775	. . . 4 |- ( ph -> Fun L )
10	9	funfnd 5112	. . 3 |- ( ph -> L Fn dom L )
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 11762	. . . . . . . . 9 |- ( ph -> H : NN0 --> ( A ^pm om ) )
12	11	ffnd 5231	. . . . . . . 8 |- ( ph -> H Fn NN0 )
13		fniunfv 5617	. . . . . . . 8 |- ( H Fn NN0 -> U_ i e. NN0 ( H ` i ) = U. ran H )
14	12, 13	syl 14	. . . . . . 7 |- ( ph -> U_ i e. NN0 ( H ` i ) = U. ran H )
15	8, 14	syl5eq 2159	. . . . . 6 |- ( ph -> L = U. ran H )
16	15	rneqd 4728	. . . . 5 |- ( ph -> ran L = ran U. ran H )
17		rnuni 4908	. . . . 5 |- ran U. ran H = U_ x e. ran H ran x
18	16, 17	syl6eq 2163	. . . 4 |- ( ph -> ran L = U_ x e. ran H ran x )
19	11	frnd 5240	. . . . . . . . . 10 |- ( ph -> ran H C_ ( A ^pm om ) )
20	19	sselda 3063	. . . . . . . . 9 |- ( ( ph /\ x e. ran H ) -> x e. ( A ^pm om ) )
21		elpmi 6515	. . . . . . . . 9 |- ( x e. ( A ^pm om ) -> ( x : dom x --> A /\ dom x C_ om ) )
22	20, 21	syl 14	. . . . . . . 8 |- ( ( ph /\ x e. ran H ) -> ( x : dom x --> A /\ dom x C_ om ) )
23	22	simpld 111	. . . . . . 7 |- ( ( ph /\ x e. ran H ) -> x : dom x --> A )
24	23	frnd 5240	. . . . . 6 |- ( ( ph /\ x e. ran H ) -> ran x C_ A )
25	24	ralrimiva 2479	. . . . 5 |- ( ph -> A. x e. ran H ran x C_ A )
26		iunss 3820	. . . . 5 |- ( U_ x e. ran H ran x C_ A <-> A. x e. ran H ran x C_ A )
27	25, 26	sylibr 133	. . . 4 |- ( ph -> U_ x e. ran H ran x C_ A )
28	18, 27	eqsstrd 3099	. . 3 |- ( ph -> ran L C_ A )
29		df-f 5085	. . 3 |- ( L : dom L --> A <-> ( L Fn dom L /\ ran L C_ A ) )
30	10, 28, 29	sylanbrc 411	. 2 |- ( ph -> L : dom L --> A )
31	19	sselda 3063	. . . . . . . 8 |- ( ( ph /\ s e. ran H ) -> s e. ( A ^pm om ) )
32		pmfun 6516	. . . . . . . 8 |- ( s e. ( A ^pm om ) -> Fun s )
33	31, 32	syl 14	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun s )
34	11	ffund 5234	. . . . . . . . . 10 |- ( ph -> Fun H )
35	34	adantr 272	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> Fun H )
36		simpr 109	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> s e. ran H )
37		elrnrexdm 5513	. . . . . . . . 9 |- ( Fun H -> ( s e. ran H -> E. q e. dom H s = ( H ` q ) ) )
38	35, 36, 37	sylc 62	. . . . . . . 8 |- ( ( ph /\ s e. ran H ) -> E. q e. dom H s = ( H ` q ) )
39	1	adantr 272	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> A. x e. A A. y e. A DECID x = y )
40	2	adantr 272	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> F : om -onto-> A )
41	3	adantr 272	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
42	11	fdmd 5237	. . . . . . . . . . . . . 14 |- ( ph -> dom H = NN0 )
43	42	eleq2d 2184	. . . . . . . . . . . . 13 |- ( ph -> ( q e. dom H <-> q e. NN0 ) )
44	43	biimpa 292	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> q e. NN0 )
45	39, 40, 41, 4, 5, 6, 7, 44	ennnfonelemhf1o 11771	. . . . . . . . . . 11 |- ( ( ph /\ q e. dom H ) -> ( H ` q ) : dom ( H ` q ) -1-1-onto-> ( F " ( `' N ` q ) ) )
46		f1ocnv 5336	. . . . . . . . . . 11 |- ( ( H ` q ) : dom ( H ` q ) -1-1-onto-> ( F " ( `' N ` q ) ) -> `' ( H ` q ) : ( F " ( `' N ` q ) ) -1-1-onto-> dom ( H ` q ) )
47		f1ofun 5325	. . . . . . . . . . 11 |- ( `' ( H ` q ) : ( F " ( `' N ` q ) ) -1-1-onto-> dom ( H ` q ) -> Fun `' ( H ` q ) )
48	45, 46, 47	3syl 17	. . . . . . . . . 10 |- ( ( ph /\ q e. dom H ) -> Fun `' ( H ` q ) )
49	48	ad2ant2r 498	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' ( H ` q ) )
50		simprr 504	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> s = ( H ` q ) )
51	50	cnveqd 4675	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> `' s = `' ( H ` q ) )
52	51	funeqd 5103	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> ( Fun `' s <-> Fun `' ( H ` q ) ) )
53	49, 52	mpbird 166	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' s )
54	38, 53	rexlimddv 2528	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun `' s )
55	1	ad2antrr 477	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> A. x e. A A. y e. A DECID x = y )
56	2	ad2antrr 477	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> F : om -onto-> A )
57	3	ad2antrr 477	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
58		simplr 502	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> s e. ran H )
59		simpr 109	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> t e. ran H )
60	55, 56, 57, 4, 5, 6, 7, 58, 59	ennnfonelemrnh 11774	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> ( s C_ t \/ t C_ s ) )
61	60	ralrimiva 2479	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> A. t e. ran H ( s C_ t \/ t C_ s ) )
62	33, 54, 61	jca31 305	. . . . . 6 |- ( ( ph /\ s e. ran H ) -> ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
63	62	ralrimiva 2479	. . . . 5 |- ( ph -> A. s e. ran H ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
64		fun11uni 5151	. . . . 5 |- ( A. s e. ran H ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) -> ( Fun U. ran H /\ Fun `' U. ran H ) )
65	63, 64	syl 14	. . . 4 |- ( ph -> ( Fun U. ran H /\ Fun `' U. ran H ) )
66	65	simprd 113	. . 3 |- ( ph -> Fun `' U. ran H )
67	15	cnveqd 4675	. . . 4 |- ( ph -> `' L = `' U. ran H )
68	67	funeqd 5103	. . 3 |- ( ph -> ( Fun `' L <-> Fun `' U. ran H ) )
69	66, 68	mpbird 166	. 2 |- ( ph -> Fun `' L )
70		df-f1 5086	. 2 |- ( L : dom L -1-1-> A <-> ( L : dom L --> A /\ Fun `' L ) )
71	30, 69, 70	sylanbrc 411	1 |- ( ph -> L : dom L -1-1-> A )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 680  DECID wdc 802   = wceq 1314   e. wcel 1463   =/= wne 2282   A.wral 2390   E.wrex 2391   u. cun 3035   C_ wss 3037   (/)c0 3329   ifcif 3440   {csn 3493   <.cop 3496   U.cuni 3702   U_ciun 3779   |-> cmpt 3949   suc csuc 4247   omcom 4464   `'ccnv 4498   dom cdm 4499   ran crn 4500   "cima 4502   Fun wfun 5075   Fn wfn 5076   -->wf 5077   -1-1->wf1 5078   -onto->wfo 5079   -1-1-onto->wf1o 5080   ` cfv 5081  (class class class)co 5728   e. cmpo 5730  freccfrec 6241   ^pm cpm 6497   0cc0 7547   1c1 7548   + caddc 7550   - cmin 7856   NN0cn0 8881   ZZcz 8958   seqcseq 10111

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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pm 6499  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-z 8959  df-uz 9229  df-seqfrec 10112

This theorem is referenced by:  ennnfonelemrn  11777  ennnfonelemen  11779