Theorem ennnfonelemrn 12470

Description: Lemma for ennnfone 12476. L is onto A. (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
ennnfonelemrn	|- ( ph -> ran L = A )

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

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

Proof of Theorem ennnfonelemrn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . 4 |- ( ph -> A. x e. A A. y e. A DECID x = y )
2		ennnfonelemh.f	. . . 4 |- ( ph -> F : om -onto-> A )
3		ennnfonelemh.ne	. . . 4 |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
4		ennnfonelemh.g	. . . 4 |- 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	. . . 4 |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
6		ennnfonelemh.j	. . . 4 |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )
7		ennnfonelemh.h	. . . 4 |- H = seq 0 ( G , J )
8		ennnfone.l	. . . 4 |- L = U_ i e. NN0 ( H ` i )
9	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemf1 12469	. . 3 |- ( ph -> L : dom L -1-1-> A )
10		f1f 5440	. . 3 |- ( L : dom L -1-1-> A -> L : dom L --> A )
11		frn 5393	. . 3 |- ( L : dom L --> A -> ran L C_ A )
12	9, 10, 11	3syl 17	. 2 |- ( ph -> ran L C_ A )
13		foelrn 5774	. . . . . 6 |- ( ( F : om -onto-> A /\ w e. A ) -> E. j e. om w = ( F ` j ) )
14	2, 13	sylan 283	. . . . 5 |- ( ( ph /\ w e. A ) -> E. j e. om w = ( F ` j ) )
15		0zd 9295	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> 0 e. ZZ )
16		simprl 529	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> j e. om )
17		peano2 4612	. . . . . . . . 9 |- ( j e. om -> suc j e. om )
18	16, 17	syl 14	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> suc j e. om )
19	15, 5, 18	frec2uzuzd 10433	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. ( ZZ>= ` 0 ) )
20		nn0uz 9592	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
21	19, 20	eleqtrrdi 2283	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. NN0 )
22		fofn 5459	. . . . . . . . . 10 |- ( F : om -onto-> A -> F Fn om )
23	2, 22	syl 14	. . . . . . . . 9 |- ( ph -> F Fn om )
24	23	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> F Fn om )
25		ordom 4624	. . . . . . . . 9 |- Ord om
26		ordsucss 4521	. . . . . . . . 9 |- ( Ord om -> ( j e. om -> suc j C_ om ) )
27	25, 16, 26	mpsyl 65	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> suc j C_ om )
28		vex 2755	. . . . . . . . . 10 |- j e. _V
29	28	sucid 4435	. . . . . . . . 9 |- j e. suc j
30	29	a1i 9	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> j e. suc j )
31		fnfvima 5772	. . . . . . . 8 |- ( ( F Fn om /\ suc j C_ om /\ j e. suc j ) -> ( F ` j ) e. ( F " suc j ) )
32	24, 27, 30, 31	syl3anc 1249	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F ` j ) e. ( F " suc j ) )
33		simprr 531	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w = ( F ` j ) )
34	15, 5	frec2uzf1od 10437	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> N : om -1-1-onto-> ( ZZ>= ` 0 ) )
35		f1ocnvfv1 5799	. . . . . . . . 9 |- ( ( N : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc j e. om ) -> ( `' N ` ( N ` suc j ) ) = suc j )
36	34, 18, 35	syl2anc 411	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( `' N ` ( N ` suc j ) ) = suc j )
37	36	imaeq2d 4988	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F " ( `' N ` ( N ` suc j ) ) ) = ( F " suc j ) )
38	32, 33, 37	3eltr4d 2273	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w e. ( F " ( `' N ` ( N ` suc j ) ) ) )
39		fveq2 5534	. . . . . . . . 9 |- ( i = ( N ` suc j ) -> ( `' N ` i ) = ( `' N ` ( N ` suc j ) ) )
40	39	imaeq2d 4988	. . . . . . . 8 |- ( i = ( N ` suc j ) -> ( F " ( `' N ` i ) ) = ( F " ( `' N ` ( N ` suc j ) ) ) )
41	40	eleq2d 2259	. . . . . . 7 |- ( i = ( N ` suc j ) -> ( w e. ( F " ( `' N ` i ) ) <-> w e. ( F " ( `' N ` ( N ` suc j ) ) ) ) )
42	41	rspcev 2856	. . . . . 6 |- ( ( ( N ` suc j ) e. NN0 /\ w e. ( F " ( `' N ` ( N ` suc j ) ) ) ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
43	21, 38, 42	syl2anc 411	. . . . 5 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
44	14, 43	rexlimddv 2612	. . . 4 |- ( ( ph /\ w e. A ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
45		eliun 3905	. . . 4 |- ( w e. U_ i e. NN0 ( F " ( `' N ` i ) ) <-> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
46	44, 45	sylibr 134	. . 3 |- ( ( ph /\ w e. A ) -> w e. U_ i e. NN0 ( F " ( `' N ` i ) ) )
47	8	rneqi 4873	. . . . . . 7 |- ran L = ran U_ i e. NN0 ( H ` i )
48		rniun 5057	. . . . . . 7 |- ran U_ i e. NN0 ( H ` i ) = U_ i e. NN0 ran ( H ` i )
49	47, 48	eqtri 2210	. . . . . 6 |- ran L = U_ i e. NN0 ran ( H ` i )
50	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> A. x e. A A. y e. A DECID x = y )
51	2	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> F : om -onto-> A )
52	3	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )
53		simpr 110	. . . . . . . . 9 |- ( ( ph /\ i e. NN0 ) -> i e. NN0 )
54	50, 51, 52, 4, 5, 6, 7, 53	ennnfonelemhf1o 12464	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) )
55		f1ofo 5487	. . . . . . . 8 |- ( ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) -> ( H ` i ) : dom ( H ` i ) -onto-> ( F " ( `' N ` i ) ) )
56		forn 5460	. . . . . . . 8 |- ( ( H ` i ) : dom ( H ` i ) -onto-> ( F " ( `' N ` i ) ) -> ran ( H ` i ) = ( F " ( `' N ` i ) ) )
57	54, 55, 56	3syl 17	. . . . . . 7 |- ( ( ph /\ i e. NN0 ) -> ran ( H ` i ) = ( F " ( `' N ` i ) ) )
58	57	iuneq2dv 3922	. . . . . 6 |- ( ph -> U_ i e. NN0 ran ( H ` i ) = U_ i e. NN0 ( F " ( `' N ` i ) ) )
59	49, 58	eqtrid 2234	. . . . 5 |- ( ph -> ran L = U_ i e. NN0 ( F " ( `' N ` i ) ) )
60	59	eleq2d 2259	. . . 4 |- ( ph -> ( w e. ran L <-> w e. U_ i e. NN0 ( F " ( `' N ` i ) ) ) )
61	60	adantr 276	. . 3 |- ( ( ph /\ w e. A ) -> ( w e. ran L <-> w e. U_ i e. NN0 ( F " ( `' N ` i ) ) ) )
62	46, 61	mpbird 167	. 2 |- ( ( ph /\ w e. A ) -> w e. ran L )
63	12, 62	eqelssd 3189	1 |- ( ph -> ran L = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2160   =/= wne 2360   A.wral 2468   E.wrex 2469   u. cun 3142   C_ wss 3144   (/)c0 3437   ifcif 3549   {csn 3607   <.cop 3610   U_ciun 3901   |-> cmpt 4079   Ord word 4380   suc csuc 4383   omcom 4607   `'ccnv 4643   dom cdm 4644   ran crn 4645   "cima 4647   Fn wfn 5230   -->wf 5231   -1-1->wf1 5232   -onto->wfo 5233   -1-1-onto->wf1o 5234   ` cfv 5235  (class class class)co 5896   e. cmpo 5898  freccfrec 6415   ^pm cpm 6675   0cc0 7841   1c1 7842   + caddc 7844   - cmin 8158   NN0cn0 9206   ZZcz 9283   ZZ>=cuz 9558   seqcseq 10476

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pm 6677  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-inn 8950  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477

This theorem is referenced by:  ennnfonelemen  12472