Theorem ennnfonelemf1 12822

Description: Lemma for ennnfone 12829. 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 12821	. . . 4 |- ( ph -> Fun L )
10	9	funfnd 5303	. . 3 |- ( ph -> L Fn dom L )
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12808	. . . . . . . . 9 |- ( ph -> H : NN0 --> ( A ^pm om ) )
12	11	ffnd 5428	. . . . . . . 8 |- ( ph -> H Fn NN0 )
13		fniunfv 5833	. . . . . . . 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 2250	. . . . . 6 |- ( ph -> L = U. ran H )
16	15	rneqd 4908	. . . . 5 |- ( ph -> ran L = ran U. ran H )
17		rnuni 5095	. . . . 5 |- ran U. ran H = U_ x e. ran H ran x
18	16, 17	eqtrdi 2254	. . . 4 |- ( ph -> ran L = U_ x e. ran H ran x )
19	11	frnd 5437	. . . . . . . . . 10 |- ( ph -> ran H C_ ( A ^pm om ) )
20	19	sselda 3193	. . . . . . . . 9 |- ( ( ph /\ x e. ran H ) -> x e. ( A ^pm om ) )
21		elpmi 6756	. . . . . . . . 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 5437	. . . . . 6 |- ( ( ph /\ x e. ran H ) -> ran x C_ A )
25	24	ralrimiva 2579	. . . . 5 |- ( ph -> A. x e. ran H ran x C_ A )
26		iunss 3968	. . . . 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 3229	. . 3 |- ( ph -> ran L C_ A )
29		df-f 5276	. . 3 |- ( L : dom L --> A <-> ( L Fn dom L /\ ran L C_ A ) )
30	10, 28, 29	sylanbrc 417	. 2 |- ( ph -> L : dom L --> A )
31	19	sselda 3193	. . . . . . . 8 |- ( ( ph /\ s e. ran H ) -> s e. ( A ^pm om ) )
32		pmfun 6757	. . . . . . . 8 |- ( s e. ( A ^pm om ) -> Fun s )
33	31, 32	syl 14	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun s )
34	11	ffund 5431	. . . . . . . . . 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 5721	. . . . . . . . 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 5434	. . . . . . . . . . . . . 14 |- ( ph -> dom H = NN0 )
43	42	eleq2d 2275	. . . . . . . . . . . . 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 12817	. . . . . . . . . . 11 |- ( ( ph /\ q e. dom H ) -> ( H ` q ) : dom ( H ` q ) -1-1-onto-> ( F " ( `' N ` q ) ) )
46		f1ocnv 5537	. . . . . . . . . . 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 5526	. . . . . . . . . . 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 509	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> Fun `' ( H ` q ) )
50		simprr 531	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> s = ( H ` q ) )
51	50	cnveqd 4855	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ran H ) /\ ( q e. dom H /\ s = ( H ` q ) ) ) -> `' s = `' ( H ` q ) )
52	51	funeqd 5294	. . . . . . . . 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 2628	. . . . . . 7 |- ( ( ph /\ s e. ran H ) -> Fun `' s )
55	1	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> A. x e. A A. y e. A DECID x = y )
56	2	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> F : om -onto-> A )
57	3	ad2antrr 488	. . . . . . . . 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 528	. . . . . . . . 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 12820	. . . . . . . 8 |- ( ( ( ph /\ s e. ran H ) /\ t e. ran H ) -> ( s C_ t \/ t C_ s ) )
61	60	ralrimiva 2579	. . . . . . 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 2579	. . . . 5 |- ( ph -> A. s e. ran H ( ( Fun s /\ Fun `' s ) /\ A. t e. ran H ( s C_ t \/ t C_ s ) ) )
64		fun11uni 5345	. . . . 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 4855	. . . 4 |- ( ph -> `' L = `' U. ran H )
68	67	funeqd 5294	. . 3 |- ( ph -> ( Fun `' L <-> Fun `' U. ran H ) )
69	66, 68	mpbird 167	. 2 |- ( ph -> Fun `' L )
70		df-f1 5277	. 2 |- ( L : dom L -1-1-> A <-> ( L : dom L --> A /\ Fun `' L ) )
71	30, 69, 70	sylanbrc 417	1 |- ( ph -> L : dom L -1-1-> A )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710  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.cuni 3850   U_ciun 3927   |-> cmpt 4106   suc csuc 4413   omcom 4639   `'ccnv 4675   dom cdm 4676   ran crn 4677   "cima 4679   Fun wfun 5266   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   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:  ennnfonelemrn  12823  ennnfonelemen  12825