Theorem ennnfonelemrn 12823

Description: Lemma for ennnfone 12829. 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 12822	. . 3 |- ( ph -> L : dom L -1-1-> A )
10		f1f 5483	. . 3 |- ( L : dom L -1-1-> A -> L : dom L --> A )
11		frn 5436	. . 3 |- ( L : dom L --> A -> ran L C_ A )
12	9, 10, 11	3syl 17	. 2 |- ( ph -> ran L C_ A )
13		foelrn 5823	. . . . . 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 9386	. . . . . . . 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 4644	. . . . . . . . 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 10549	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. ( ZZ>= ` 0 ) )
20		nn0uz 9685	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
21	19, 20	eleqtrrdi 2299	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( N ` suc j ) e. NN0 )
22		fofn 5502	. . . . . . . . . 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 4656	. . . . . . . . 9 |- Ord om
26		ordsucss 4553	. . . . . . . . 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 2775	. . . . . . . . . 10 |- j e. _V
29	28	sucid 4465	. . . . . . . . 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 5821	. . . . . . . 8 |- ( ( F Fn om /\ suc j C_ om /\ j e. suc j ) -> ( F ` j ) e. ( F " suc j ) )
32	24, 27, 30, 31	syl3anc 1250	. . . . . . 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 10553	. . . . . . . . 9 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> N : om -1-1-onto-> ( ZZ>= ` 0 ) )
35		f1ocnvfv1 5848	. . . . . . . . 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 5023	. . . . . . 7 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> ( F " ( `' N ` ( N ` suc j ) ) ) = ( F " suc j ) )
38	32, 33, 37	3eltr4d 2289	. . . . . 6 |- ( ( ( ph /\ w e. A ) /\ ( j e. om /\ w = ( F ` j ) ) ) -> w e. ( F " ( `' N ` ( N ` suc j ) ) ) )
39		fveq2 5578	. . . . . . . . 9 |- ( i = ( N ` suc j ) -> ( `' N ` i ) = ( `' N ` ( N ` suc j ) ) )
40	39	imaeq2d 5023	. . . . . . . 8 |- ( i = ( N ` suc j ) -> ( F " ( `' N ` i ) ) = ( F " ( `' N ` ( N ` suc j ) ) ) )
41	40	eleq2d 2275	. . . . . . 7 |- ( i = ( N ` suc j ) -> ( w e. ( F " ( `' N ` i ) ) <-> w e. ( F " ( `' N ` ( N ` suc j ) ) ) ) )
42	41	rspcev 2877	. . . . . 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 2628	. . . 4 |- ( ( ph /\ w e. A ) -> E. i e. NN0 w e. ( F " ( `' N ` i ) ) )
45		eliun 3931	. . . 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 4907	. . . . . . 7 |- ran L = ran U_ i e. NN0 ( H ` i )
48		rniun 5094	. . . . . . 7 |- ran U_ i e. NN0 ( H ` i ) = U_ i e. NN0 ran ( H ` i )
49	47, 48	eqtri 2226	. . . . . 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 12817	. . . . . . . 8 |- ( ( ph /\ i e. NN0 ) -> ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) )
55		f1ofo 5531	. . . . . . . 8 |- ( ( H ` i ) : dom ( H ` i ) -1-1-onto-> ( F " ( `' N ` i ) ) -> ( H ` i ) : dom ( H ` i ) -onto-> ( F " ( `' N ` i ) ) )
56		forn 5503	. . . . . . . 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 3948	. . . . . 6 |- ( ph -> U_ i e. NN0 ran ( H ` i ) = U_ i e. NN0 ( F " ( `' N ` i ) ) )
59	49, 58	eqtrid 2250	. . . . 5 |- ( ph -> ran L = U_ i e. NN0 ( F " ( `' N ` i ) ) )
60	59	eleq2d 2275	. . . 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 3212	1 |- ( ph -> ran L = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 836   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   E.wrex 2485   u. cun 3164   C_ wss 3166   (/)c0 3460   ifcif 3571   {csn 3633   <.cop 3636   U_ciun 3927   |-> cmpt 4106   Ord word 4410   suc csuc 4413   omcom 4639   `'ccnv 4675   dom cdm 4676   ran crn 4677   "cima 4679   Fn wfn 5267   -->wf 5268   -1-1->wf1 5269   -onto->wfo 5270   -1-1-onto->wf1o 5271   ` cfv 5272  (class class class)co 5946   e. cmpo 5948  freccfrec 6478   ^pm cpm 6738   0cc0 7927   1c1 7928   + caddc 7930   - cmin 8245   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   seqcseq 10594

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pm 6740  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-seqfrec 10595

This theorem is referenced by:  ennnfonelemen  12825