Theorem enumctlemm 7419

Description: Lemma for enumct 7420. 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 5596	. . . . . . 7 |- ( F : N -onto-> A -> F : N --> A )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> F : N --> A )
4	3	ffvelcdmda 5818	. . . . 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 5819	. . . . 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 6747	. . . . 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 3657	. . 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 5837	. 2 |- ( ph -> G : om --> A )
17		foelrn 5932	. . . . . 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 2295	. . . . . . . . . . 11 |- ( k = x -> ( k e. N <-> x e. N ) )
20		fveq2 5676	. . . . . . . . . . 11 |- ( k = x -> ( F ` k ) = ( F ` x ) )
21	19, 20	ifbieq1d 3650	. . . . . . . . . 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 4734	. . . . . . . . . . 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 3634	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> if ( x e. N , ( F ` x ) , ( F ` (/) ) ) = ( F ` x ) )
27	3	ffvelcdmda 5818	. . . . . . . . . . 11 |- ( ( ph /\ x e. N ) -> ( F ` x ) e. A )
28	26, 27	eqeltrd 2311	. . . . . . . . . 10 |- ( ( ph /\ x e. N ) -> if ( x e. N , ( F ` x ) , ( F ` (/) ) ) e. A )
29	15, 21, 25, 28	fvmptd3 5777	. . . . . . . . 9 |- ( ( ph /\ x e. N ) -> ( G ` x ) = if ( x e. N , ( F ` x ) , ( F ` (/) ) ) )
30	29, 26	eqtrd 2267	. . . . . . . 8 |- ( ( ph /\ x e. N ) -> ( G ` x ) = ( F ` x ) )
31	30	eqeq2d 2246	. . . . . . 7 |- ( ( ph /\ x e. N ) -> ( y = ( G ` x ) <-> y = ( F ` x ) ) )
32	31	rexbidva 2541	. . . . . 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 4737	. . . . . . 7 |- om e. On
36	35	onelssi 4556	. . . . . 6 |- ( N e. om -> N C_ om )
37		ssrexv 3307	. . . . . 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 2617	. 2 |- ( ph -> A. y e. A E. x e. om y = ( G ` x ) )
42		dffo3 5830	. 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 842   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3214   (/)c0 3512   ifcif 3625   |-> cmpt 4177   omcom 4718   -->wf 5354   -onto->wfo 5356   ` cfv 5358

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366

This theorem is referenced by:  enumct  7420