Theorem enumctlemm 7269

Description: Lemma for enumct 7270. The case where N is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)

Hypotheses

Ref	Expression
enumctlemm.f	|- ( ph -> F : N -onto-> A )
enumctlemm.n	|- ( ph -> N e. om )
enumctlemm.n0	|- ( ph -> (/) e. N )
enumctlemm.g	|- G = ( k e. om |-> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) )

Assertion

Ref	Expression
enumctlemm	|- ( ph -> G : om -onto-> A )

Distinct variable groups:   A, k   k, F   k, N   ph, k

Allowed substitution hint:   G( k)

Proof of Theorem enumctlemm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enumctlemm.f	. . . . . . 7 |- ( ph -> F : N -onto-> A )
2		fof 5544	. . . . . . 7 |- ( F : N -onto-> A -> F : N --> A )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> F : N --> A )
4	3	ffvelcdmda 5763	. . . . 5 |- ( ( ph /\ k e. N ) -> ( F ` k ) e. A )
5	4	adantlr 477	. . . 4 |- ( ( ( ph /\ k e. om ) /\ k e. N ) -> ( F ` k ) e. A )
6		enumctlemm.n0	. . . . . 6 |- ( ph -> (/) e. N )
7	3, 6	ffvelcdmd 5764	. . . . 5 |- ( ph -> ( F ` (/) ) e. A )
8	7	ad2antrr 488	. . . 4 |- ( ( ( ph /\ k e. om ) /\ -. k e. N ) -> ( F ` (/) ) e. A )
9		simpr 110	. . . . 5 |- ( ( ph /\ k e. om ) -> k e. om )
10		enumctlemm.n	. . . . . 6 |- ( ph -> N e. om )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ k e. om ) -> N e. om )
12		nndcel 6636	. . . . 5 |- ( ( k e. om /\ N e. om ) -> DECID k e. N )
13	9, 11, 12	syl2anc 411	. . . 4 |- ( ( ph /\ k e. om ) -> DECID k e. N )
14	5, 8, 13	ifcldadc 3632	. . 3 |- ( ( ph /\ k e. om ) -> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) e. A )
15		enumctlemm.g	. . 3 |- G = ( k e. om |-> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) )
16	14, 15	fmptd 5782	. 2 |- ( ph -> G : om --> A )
17		foelrn 5869	. . . . . 6 |- ( ( F : N -onto-> A /\ y e. A ) -> E. x e. N y = ( F ` x ) )
18	1, 17	sylan 283	. . . . 5 |- ( ( ph /\ y e. A ) -> E. x e. N y = ( F ` x ) )
19		eleq1w 2290	. . . . . . . . . . 11 |- ( k = x -> ( k e. N <-> x e. N ) )
20		fveq2 5623	. . . . . . . . . . 11 |- ( k = x -> ( F ` k ) = ( F ` x ) )
21	19, 20	ifbieq1d 3625	. . . . . . . . . 10 |- ( k = x -> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) = if ( x e. N , ( F ` x ) , ( F ` (/) ) ) )
22		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> x e. N )
23	10	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> N e. om )
24		elnn 4695	. . . . . . . . . . 11 |- ( ( x e. N /\ N e. om ) -> x e. om )
25	22, 23, 24	syl2anc 411	. . . . . . . . . 10 |- ( ( ph /\ x e. N ) -> x e. om )
26	22	iftrued 3609	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> if ( x e. N , ( F ` x ) , ( F ` (/) ) ) = ( F ` x ) )
27	3	ffvelcdmda 5763	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> ( F ` x ) e. A )
28	26, 27	eqeltrd 2306	. . . . . . . . . 10 |- ( ( ph /\ x e. N ) -> if ( x e. N , ( F ` x ) , ( F ` (/) ) ) e. A )
29	15, 21, 25, 28	fvmptd3 5721	. . . . . . . . 9 |- ( ( ph /\ x e. N ) -> ( G ` x ) = if ( x e. N , ( F ` x ) , ( F ` (/) ) ) )
30	29, 26	eqtrd 2262	. . . . . . . 8 |- ( ( ph /\ x e. N ) -> ( G ` x ) = ( F ` x ) )
31	30	eqeq2d 2241	. . . . . . 7 |- ( ( ph /\ x e. N ) -> ( y = ( G ` x ) <-> y = ( F ` x ) ) )
32	31	rexbidva 2527	. . . . . 6 |- ( ph -> ( E. x e. N y = ( G ` x ) <-> E. x e. N y = ( F ` x ) ) )
33	32	adantr 276	. . . . 5 |- ( ( ph /\ y e. A ) -> ( E. x e. N y = ( G ` x ) <-> E. x e. N y = ( F ` x ) ) )
34	18, 33	mpbird 167	. . . 4 |- ( ( ph /\ y e. A ) -> E. x e. N y = ( G ` x ) )
35		omelon 4698	. . . . . . 7 |- om e. On
36	35	onelssi 4517	. . . . . 6 |- ( N e. om -> N C_ om )
37		ssrexv 3289	. . . . . 6 |- ( N C_ om -> ( E. x e. N y = ( G ` x ) -> E. x e. om y = ( G ` x ) ) )
38	10, 36, 37	3syl 17	. . . . 5 |- ( ph -> ( E. x e. N y = ( G ` x ) -> E. x e. om y = ( G ` x ) ) )
39	38	adantr 276	. . . 4 |- ( ( ph /\ y e. A ) -> ( E. x e. N y = ( G ` x ) -> E. x e. om y = ( G ` x ) ) )
40	34, 39	mpd 13	. . 3 |- ( ( ph /\ y e. A ) -> E. x e. om y = ( G ` x ) )
41	40	ralrimiva 2603	. 2 |- ( ph -> A. y e. A E. x e. om y = ( G ` x ) )
42		dffo3 5775	. 2 |- ( G : om -onto-> A <-> ( G : om --> A /\ A. y e. A E. x e. om y = ( G ` x ) ) )
43	16, 41, 42	sylanbrc 417	1 |- ( ph -> G : om -onto-> A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3197   (/)c0 3491   ifcif 3602   |-> cmpt 4144   omcom 4679   -->wf 5310   -onto->wfo 5312   ` cfv 5314

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fo 5320  df-fv 5322

This theorem is referenced by:  enumct  7270