Theorem ennnfonelemrn 13310

Description: Lemma for ennnfone 13316. 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 13309	. . 3 |- ( ph -> L : dom L -1-1-> A )
10		f1f 5598	. . 3 |- ( L : dom L -1-1-> A -> L : dom L --> A )
11		frn 5542	. . 3 |- ( L : dom L --> A -> ran L C_ A )
12	9, 10, 11	3syl 17	. 2 |- ( ph -> ran L C_ A )
13		foelrn 5958	. . . . . 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 9656	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> 0 e. ZZ )
16		simprl 535	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> j e. om )
17		peano2 4742	. . . . . . . . 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 10839	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. ( ZZ>= ` 0 ) )
20		nn0uz 9957	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
21	19, 20	eleqtrrdi 2332	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. NN0 )
22		fofn 5617	. . . . . . . . . 10 |- ( F : om -onto-> A -> F Fn om )
23	2, 22	syl 14	. . . . . . . . 9 |- ( ph -> F Fn om )
24	23	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> F Fn om )
25		ordom 4754	. . . . . . . . 9 |- Ord om
26		ordsucss 4651	. . . . . . . . 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 2824	. . . . . . . . . 10 |- j e. _V
29	28	sucid 4562	. . . . . . . . 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 5953	. . . . . . . 8 |- ( ( F Fn om /\ suc j C_ om /\ j e. suc j ) -> ( F ` j ) e. ( F " suc j ) )
32	24, 27, 30, 31	syl3anc 1278	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F ` j ) e. ( F " suc j ) )
33		simprr 537	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w = ( F ` j ) )
34	15, 5	frec2uzf1od 10843	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> N : om -1-1-onto-> ( ZZ>= ` 0 ) )
35		f1ocnvfv1 5983	. . . . . . . . 9 |- ( ( N : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc j e. om ) -> ( `' N ` ( N ` suc j ) ) = suc j )
36	34, 18, 35	syl2anc 415	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( `' N ` ( N ` suc j ) ) = suc j )
37	36	imaeq2d 5126	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F " ( `' N ` ( N ` suc j ) ) ) = ( F " suc j ) )
38	32, 33, 37	3eltr4d 2322	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w e. ( F " ( `' N ` ( N ` suc j ) ) ) )
39		fveq2 5695	. . . . . . . . 9 |- ( i = ( N ` suc j ) -> ( `' N ` i ) = ( `' N ` ( N ` suc j ) ) )
40	39	imaeq2d 5126	. . . . . . . 8 |- ( i = ( N ` suc j ) -> ( F " ( `' N ` i ) ) = ( F " ( `' N ` ( N ` suc j ) ) ) )
41	40	eleq2d 2308	. . . . . . 7 |- ( i = ( N ` suc j ) -> ( w e. ( F " ( `' N ` i ) ) <-> w e. ( F " ( `' N ` ( N ` suc j ) ) ) ) )
42	41	rspcev 2929	. . . . . 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 415	. . . . 5 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
44	14, 43	rexlimddv 2673	. . . 4 |- ( ( ph /\ w e. A ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
45		eliun 4016	. . . 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 5010	. . . . . . 7 |- ran L = ran U_ i e. NN0 ( H ` i )
48		rniun 5198	. . . . . . 7 |- ran U_ i e. NN0 ( H ` i ) = U_ i e. NN0 ran ( H ` i )
49	47, 48	eqtri 2259	. . . . . 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 13304	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) )
55		f1ofo 5646	. . . . . . . 8 |- ( ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) -> ( H ` i ) : dom ( H ` i ) -onto-> ( F " ( `' N ` i ) ) )
56		forn 5618	. . . . . . . 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 4033	. . . . . 6 |- ( ph -> U_ i e. NN0 ran ( H ` i ) = U_ i e. NN0 ( F " ( `' N ` i ) ) )
59	49, 58	eqtrid 2283	. . . . 5 |- ( ph -> ran L = U_ i e. NN0 ( F " ( `' N ` i ) ) )
60	59	eleq2d 2308	. . . 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 3267	1 |- ( ph -> ran L = A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  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_ciun 4012   |-> cmpt 4192   Ord word 4507   suc csuc 4510   omcom 4737   `'ccnv 4773   dom cdm 4774   ran crn 4775   "cima 4777   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   ZZ>=cuz 9921   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:  ennnfonelemen  13312