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Theorem enumctlemm 7115
Description: Lemma for enumct 7116. 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.
StepHypRef Expression
1 enumctlemm.f . . . . . . 7  |-  ( ph  ->  F : N -onto-> A
)
2 fof 5440 . . . . . . 7  |-  ( F : N -onto-> A  ->  F : N --> A )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  F : N --> A )
43ffvelcdmda 5653 . . . . 5  |-  ( (
ph  /\  k  e.  N )  ->  ( F `  k )  e.  A )
54adantlr 477 . . . 4  |-  ( ( ( ph  /\  k  e.  om )  /\  k  e.  N )  ->  ( F `  k )  e.  A )
6 enumctlemm.n0 . . . . . 6  |-  ( ph  -> 
(/)  e.  N )
73, 6ffvelcdmd 5654 . . . . 5  |-  ( ph  ->  ( F `  (/) )  e.  A )
87ad2antrr 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 )
1110adantr 276 . . . . 5  |-  ( (
ph  /\  k  e.  om )  ->  N  e.  om )
12 nndcel 6503 . . . . 5  |-  ( ( k  e.  om  /\  N  e.  om )  -> DECID  k  e.  N )
139, 11, 12syl2anc 411 . . . 4  |-  ( (
ph  /\  k  e.  om )  -> DECID  k  e.  N
)
145, 8, 13ifcldadc 3565 . . 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 `  (/) ) ) )
1614, 15fmptd 5672 . 2  |-  ( ph  ->  G : om --> A )
17 foelrn 5755 . . . . . 6  |-  ( ( F : N -onto-> A  /\  y  e.  A
)  ->  E. x  e.  N  y  =  ( F `  x ) )
181, 17sylan 283 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  E. x  e.  N  y  =  ( F `  x ) )
19 eleq1w 2238 . . . . . . . . . . 11  |-  ( k  =  x  ->  (
k  e.  N  <->  x  e.  N ) )
20 fveq2 5517 . . . . . . . . . . 11  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2119, 20ifbieq1d 3558 . . . . . . . . . 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 )
2310adantr 276 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  N  e.  om )
24 elnn 4607 . . . . . . . . . . 11  |-  ( ( x  e.  N  /\  N  e.  om )  ->  x  e.  om )
2522, 23, 24syl2anc 411 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  x  e.  om )
2622iftrued 3543 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  =  ( F `  x ) )
273ffvelcdmda 5653 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  ( F `  x )  e.  A )
2826, 27eqeltrd 2254 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  e.  A )
2915, 21, 25, 28fvmptd3 5611 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  if ( x  e.  N ,  ( F `
 x ) ,  ( F `  (/) ) ) )
3029, 26eqtrd 2210 . . . . . . . 8  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  ( F `  x ) )
3130eqeq2d 2189 . . . . . . 7  |-  ( (
ph  /\  x  e.  N )  ->  (
y  =  ( G `
 x )  <->  y  =  ( F `  x ) ) )
3231rexbidva 2474 . . . . . 6  |-  ( ph  ->  ( E. x  e.  N  y  =  ( G `  x )  <->  E. x  e.  N  y  =  ( F `  x ) ) )
3332adantr 276 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  ( E. x  e.  N  y  =  ( G `  x )  <->  E. x  e.  N  y  =  ( F `  x ) ) )
3418, 33mpbird 167 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  E. x  e.  N  y  =  ( G `  x ) )
35 omelon 4610 . . . . . . 7  |-  om  e.  On
3635onelssi 4431 . . . . . 6  |-  ( N  e.  om  ->  N  C_ 
om )
37 ssrexv 3222 . . . . . 6  |-  ( N 
C_  om  ->  ( E. x  e.  N  y  =  ( G `  x )  ->  E. x  e.  om  y  =  ( G `  x ) ) )
3810, 36, 373syl 17 . . . . 5  |-  ( ph  ->  ( E. x  e.  N  y  =  ( G `  x )  ->  E. x  e.  om  y  =  ( G `  x ) ) )
3938adantr 276 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( E. x  e.  N  y  =  ( G `  x )  ->  E. x  e.  om  y  =  ( G `  x ) ) )
4034, 39mpd 13 . . 3  |-  ( (
ph  /\  y  e.  A )  ->  E. x  e.  om  y  =  ( G `  x ) )
4140ralrimiva 2550 . 2  |-  ( ph  ->  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) )
42 dffo3 5665 . 2  |-  ( G : om -onto-> A  <->  ( G : om --> A  /\  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) ) )
4316, 41, 42sylanbrc 417 1  |-  ( ph  ->  G : om -onto-> A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105  DECID wdc 834    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3131   (/)c0 3424   ifcif 3536    |-> cmpt 4066   omcom 4591   -->wf 5214   -onto->wfo 5216   ` cfv 5218
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226
This theorem is referenced by:  enumct  7116
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