Theorem ennnfonelemf1 12288

Description: Lemma for ennnfone 12295. 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 12287	. . . 4 |- ( ph -> Fun L )
10	9	funfnd 5213	. . 3 |- ( ph -> L Fn dom L )
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12274	. . . . . . . . 9 |- ( ph -> H : NN0 --> ( A ^pm om ) )
12	11	ffnd 5332	. . . . . . . 8 |- ( ph -> H Fn NN0 )
13		fniunfv 5724	. . . . . . . 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 2209	. . . . . 6 |- ( ph -> L = U. ran H )
16	15	rneqd 4827	. . . . 5 |- ( ph -> ran L = ran U. ran H )
17		rnuni 5009	. . . . 5 |- ran U. ran H = U_ x e. ran H ran x
18	16, 17	eqtrdi 2213	. . . 4 |- ( ph -> ran L = U_ x e. ran H ran x )
19	11	frnd 5341	. . . . . . . . . 10 |- ( ph -> ran H C_ ( A ^pm om ) )
20	19	sselda 3137	. . . . . . . . 9 |- ( ( ph /\ x e. ran H ) -> x e. ( A ^pm om ) )
21		elpmi 6624	. . . . . . . . 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 5341	. . . . . 6 |- ( ( ph /\ x e. ran H ) -> ran x C_ A )
25	24	ralrimiva 2537	. . . . 5 |- ( ph -> A. x e. ran H ran x C_ A )
26		iunss 3901	. . . . 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 3173	. . 3 |- ( ph -> ran L C_ A )
29		df-f 5186	. . 3 |- ( L : dom L --> A <-> ( L Fn dom L /\ ran L C_ A ) )
30	10, 28, 29	sylanbrc 414	. 2 |- ( ph -> L : dom L --> A )
31	19	sselda 3137	. . . . . . . 8 |- ( ( ph /\ s e. ran H ) -> s e. ( A ^pm om ) )
32		pmfun 6625	. . . . . . . 8 |- ( s e. ( A ^pm om ) -> Fun s )
33	31, 32	syl 14	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun s )
34	11	ffund 5335	. . . . . . . . . 10 |- ( ph -> Fun H )
35	34	adantr 274	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> Fun H )
36		simpr 109	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> s e. ran H )
37		elrnrexdm 5618	. . . . . . . . 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 274	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> A. x e. A A. y e. A DECID x = y )
40	2	adantr 274	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> F : om -onto-> A )
41	3	adantr 274	. . . . . . . . . . . 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 5338	. . . . . . . . . . . . . 14 |- ( ph -> dom H = NN0 )
43	42	eleq2d 2234	. . . . . . . . . . . . 13 |- ( ph -> ( q e. dom H <-> q e. NN0 ) )
44	43	biimpa 294	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> q e. NN0 )
45	39, 40, 41, 4, 5, 6, 7, 44	ennnfonelemhf1o 12283	. . . . . . . . . . 11 |- ( ( ph /\ q e. dom H ) -> ( H ` q ) : dom ( H ` q ) -1-1-onto-> ( F " ( `' N ` q ) ) )
46		f1ocnv 5439	. . . . . . . . . . 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 5428	. . . . . . . . . . 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 501	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' ( H ` q ) )
50		simprr 522	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> s = ( H ` q ) )
51	50	cnveqd 4774	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> `' s = `' ( H ` q ) )
52	51	funeqd 5204	. . . . . . . . 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 2586	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun `' s )
55	1	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> A. x e. A A. y e. A DECID x = y )
56	2	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> F : om -onto-> A )
57	3	ad2antrr 480	. . . . . . . . 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 520	. . . . . . . . 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 12286	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> ( s C_ t \/ t C_ s ) )
61	60	ralrimiva 2537	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> A. t e. ran H ( s C_ t \/ t C_ s ) )
62	33, 54, 61	jca31 307	. . . . . 6 |- ( ( ph /\ s e. ran H ) -> ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
63	62	ralrimiva 2537	. . . . 5 |- ( ph -> A. s e. ran H ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
64		fun11uni 5252	. . . . 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 4774	. . . 4 |- ( ph -> `' L = `' U. ran H )
68	67	funeqd 5204	. . 3 |- ( ph -> ( Fun `' L <-> Fun `' U. ran H ) )
69	66, 68	mpbird 166	. 2 |- ( ph -> Fun `' L )
70		df-f1 5187	. 2 |- ( L : dom L -1-1-> A <-> ( L : dom L --> A /\ Fun `' L ) )
71	30, 69, 70	sylanbrc 414	1 |- ( ph -> L : dom L -1-1-> A )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698  DECID wdc 824   = wceq 1342   e. wcel 2135   =/= wne 2334   A.wral 2442   E.wrex 2443   u. cun 3109   C_ wss 3111   (/)c0 3404   ifcif 3515   {csn 3570   <.cop 3573   U.cuni 3783   U_ciun 3860   |-> cmpt 4037   suc csuc 4337   omcom 4561   `'ccnv 4597   dom cdm 4598   ran crn 4599   "cima 4601   Fun wfun 5176   Fn wfn 5177   -->wf 5178   -1-1->wf1 5179   -onto->wfo 5180   -1-1-onto->wf1o 5181   ` cfv 5182  (class class class)co 5836   e. cmpo 5838  freccfrec 6349   ^pm cpm 6606   0cc0 7744   1c1 7745   + caddc 7747   - cmin 8060   NN0cn0 9105   ZZcz 9182   seqcseq 10370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pm 6608  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-seqfrec 10371

This theorem is referenced by:  ennnfonelemrn  12289  ennnfonelemen  12291