Theorem ennnfonelemf1 13309

Description: Lemma for ennnfone 13316. 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 13308	. . . 4 |- ( ph -> Fun L )
10	9	funfnd 5408	. . 3 |- ( ph -> L Fn dom L )
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 13295	. . . . . . . . 9 |- ( ph -> H : NN0 --> ( A ^pm om ) )
12	11	ffnd 5534	. . . . . . . 8 |- ( ph -> H Fn NN0 )
13		fniunfv 5968	. . . . . . . 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	eqtrid 2283	. . . . . 6 |- ( ph -> L = U. ran H )
16	15	rneqd 5011	. . . . 5 |- ( ph -> ran L = ran U. ran H )
17		rnuni 5199	. . . . 5 |- ran U. ran H = U_ x e. ran H ran x
18	16, 17	eqtrdi 2287	. . . 4 |- ( ph -> ran L = U_ x e. ran H ran x )
19	11	frnd 5543	. . . . . . . . . 10 |- ( ph -> ran H C_ ( A ^pm om ) )
20	19	sselda 3248	. . . . . . . . 9 |- ( ( ph /\ x e. ran H ) -> x e. ( A ^pm om ) )
21		elpmi 6941	. . . . . . . . 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 112	. . . . . . 7 |- ( ( ph /\ x e. ran H ) -> x : dom x --> A )
24	23	frnd 5543	. . . . . 6 |- ( ( ph /\ x e. ran H ) -> ran x C_ A )
25	24	ralrimiva 2623	. . . . 5 |- ( ph -> A. x e. ran H ran x C_ A )
26		iunss 4053	. . . . 5 |- ( U_ x e. ran H ran x C_ A <-> A. x e. ran H ran x C_ A )
27	25, 26	sylibr 134	. . . 4 |- ( ph -> U_ x e. ran H ran x C_ A )
28	18, 27	eqsstrd 3284	. . 3 |- ( ph -> ran L C_ A )
29		df-f 5381	. . 3 |- ( L : dom L --> A <-> ( L Fn dom L /\ ran L C_ A ) )
30	10, 28, 29	sylanbrc 421	. 2 |- ( ph -> L : dom L --> A )
31	19	sselda 3248	. . . . . . . 8 |- ( ( ph /\ s e. ran H ) -> s e. ( A ^pm om ) )
32		pmfun 6942	. . . . . . . 8 |- ( s e. ( A ^pm om ) -> Fun s )
33	31, 32	syl 14	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun s )
34	11	ffund 5537	. . . . . . . . . 10 |- ( ph -> Fun H )
35	34	adantr 276	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> Fun H )
36		simpr 110	. . . . . . . . 9 |- ( ( ph /\ s e. ran H ) -> s e. ran H )
37		elrnrexdm 5847	. . . . . . . . 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 276	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> A. x e. A A. y e. A DECID x = y )
40	2	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> F : om -onto-> A )
41	3	adantr 276	. . . . . . . . . . . 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 5540	. . . . . . . . . . . . . 14 |- ( ph -> dom H = NN0 )
43	42	eleq2d 2308	. . . . . . . . . . . . 13 |- ( ph -> ( q e. dom H <-> q e. NN0 ) )
44	43	biimpa 296	. . . . . . . . . . . 12 |- ( ( ph /\ q e. dom H ) -> q e. NN0 )
45	39, 40, 41, 4, 5, 6, 7, 44	ennnfonelemhf1o 13304	. . . . . . . . . . 11 |- ( ( ph /\ q e. dom H ) -> ( H ` q ) : dom ( H ` q ) -1-1-onto-> ( F " ( `' N ` q ) ) )
46		f1ocnv 5652	. . . . . . . . . . 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 5641	. . . . . . . . . . 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 513	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' ( H ` q ) )
50		simprr 537	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> s = ( H ` q ) )
51	50	cnveqd 4956	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> `' s = `' ( H ` q ) )
52	51	funeqd 5399	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> ( Fun `' s <-> Fun `' ( H ` q ) ) )
53	49, 52	mpbird 167	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' s )
54	38, 53	rexlimddv 2673	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun `' s )
55	1	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> A. x e. A A. y e. A DECID x = y )
56	2	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> F : om -onto-> A )
57	3	ad2antrr 492	. . . . . . . . 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 533	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> s e. ran H )
59		simpr 110	. . . . . . . . 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 13307	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> ( s C_ t \/ t C_ s ) )
61	60	ralrimiva 2623	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> A. t e. ran H ( s C_ t \/ t C_ s ) )
62	33, 54, 61	jca31 309	. . . . . 6 |- ( ( ph /\ s e. ran H ) -> ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
63	62	ralrimiva 2623	. . . . 5 |- ( ph -> A. s e. ran H ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
64		fun11uni 5451	. . . . 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 114	. . 3 |- ( ph -> Fun `' U. ran H )
67	15	cnveqd 4956	. . . 4 |- ( ph -> `' L = `' U. ran H )
68	67	funeqd 5399	. . 3 |- ( ph -> ( Fun `' L <-> Fun `' U. ran H ) )
69	66, 68	mpbird 167	. 2 |- ( ph -> Fun `' L )
70		df-f1 5382	. 2 |- ( L : dom L -1-1-> A <-> ( L : dom L --> A /\ Fun `' L ) )
71	30, 69, 70	sylanbrc 421	1 |- ( ph -> L : dom L -1-1-> A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   u. cun 3218   C_ wss 3220   (/)c0 3520   ifcif 3638   {csn 3709   <.cop 3712   U.cuni 3935   U_ciun 4012   |-> cmpt 4192   suc csuc 4510   omcom 4737   `'ccnv 4773   dom cdm 4774   ran crn 4775   "cima 4777   Fun wfun 5371   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   -onto->wfo 5375   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   e. cmpo 6087  freccfrec 6661   ^pm cpm 6923   0cc0 8179   1c1 8180   + caddc 8182   - cmin 8497   NN0cn0 9563   ZZcz 9644   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pm 6925  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-seqfrec 10885

This theorem is used by:  ennnfonelemrn  13310  ennnfonelemen  13312