Theorem ennnfonelemrn 13120

Description: Lemma for ennnfone 13126. 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 13119	. . 3 |- ( ph -> L : dom L -1-1-> A )
10		f1f 5551	. . 3 |- ( L : dom L -1-1-> A -> L : dom L --> A )
11		frn 5498	. . 3 |- ( L : dom L --> A -> ran L C_ A )
12	9, 10, 11	3syl 17	. 2 |- ( ph -> ran L C_ A )
13		foelrn 5903	. . . . . 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 9552	. . . . . . . 8 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> 0 e. ZZ )
16		simprl 531	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> j e. om )
17		peano2 4699	. . . . . . . . 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 10727	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. ( ZZ>= ` 0 ) )
20		nn0uz 9852	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
21	19, 20	eleqtrrdi 2325	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. NN0 )
22		fofn 5570	. . . . . . . . . 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 4711	. . . . . . . . 9 |- Ord om
26		ordsucss 4608	. . . . . . . . 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 2806	. . . . . . . . . 10 |- j e. _V
29	28	sucid 4520	. . . . . . . . 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 5899	. . . . . . . 8 |- ( ( F Fn om /\ suc j C_ om /\ j e. suc j ) -> ( F ` j ) e. ( F " suc j ) )
32	24, 27, 30, 31	syl3anc 1274	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F ` j ) e. ( F " suc j ) )
33		simprr 533	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w = ( F ` j ) )
34	15, 5	frec2uzf1od 10731	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> N : om -1-1-onto-> ( ZZ>= ` 0 ) )
35		f1ocnvfv1 5928	. . . . . . . . 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 5082	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F " ( `' N ` ( N ` suc j ) ) ) = ( F " suc j ) )
38	32, 33, 37	3eltr4d 2315	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w e. ( F " ( `' N ` ( N ` suc j ) ) ) )
39		fveq2 5648	. . . . . . . . 9 |- ( i = ( N ` suc j ) -> ( `' N ` i ) = ( `' N ` ( N ` suc j ) ) )
40	39	imaeq2d 5082	. . . . . . . 8 |- ( i = ( N ` suc j ) -> ( F " ( `' N ` i ) ) = ( F " ( `' N ` ( N ` suc j ) ) ) )
41	40	eleq2d 2301	. . . . . . 7 |- ( i = ( N ` suc j ) -> ( w e. ( F " ( `' N ` i ) ) <-> w e. ( F " ( `' N ` ( N ` suc j ) ) ) ) )
42	41	rspcev 2911	. . . . . 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 2656	. . . 4 |- ( ( ph /\ w e. A ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
45		eliun 3979	. . . 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 4966	. . . . . . 7 |- ran L = ran U_ i e. NN0 ( H ` i )
48		rniun 5154	. . . . . . 7 |- ran U_ i e. NN0 ( H ` i ) = U_ i e. NN0 ran ( H ` i )
49	47, 48	eqtri 2252	. . . . . 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 13114	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) )
55		f1ofo 5599	. . . . . . . 8 |- ( ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) -> ( H ` i ) : dom ( H ` i ) -onto-> ( F " ( `' N ` i ) ) )
56		forn 5571	. . . . . . . 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 3996	. . . . . 6 |- ( ph -> U_ i e. NN0 ran ( H ` i ) = U_ i e. NN0 ( F " ( `' N ` i ) ) )
59	49, 58	eqtrid 2276	. . . . 5 |- ( ph -> ran L = U_ i e. NN0 ( F " ( `' N ` i ) ) )
60	59	eleq2d 2301	. . . 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 3247	1 |- ( ph -> ran L = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   E.wrex 2512   u. cun 3199   C_ wss 3201   (/)c0 3496   ifcif 3607   {csn 3673   <.cop 3676   U_ciun 3975   |-> cmpt 4155   Ord word 4465   suc csuc 4468   omcom 4694   `'ccnv 4730   dom cdm 4731   ran crn 4732   "cima 4734   Fn wfn 5328   -->wf 5329   -1-1->wf1 5330   -onto->wfo 5331   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   e. cmpo 6030  freccfrec 6599   ^pm cpm 6861   0cc0 8092   1c1 8093   + caddc 8095   - cmin 8409   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   seqcseq 10772

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pm 6863  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773

This theorem is referenced by:  ennnfonelemen  13122