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Theorem enumctlemm 7126
Description: Lemma for enumct 7127. 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 5450 . . . . . . 7  |-  ( F : N -onto-> A  ->  F : N --> A )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  F : N --> A )
43ffvelcdmda 5664 . . . . 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 5665 . . . . 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 6514 . . . . 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 3575 . . 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 5683 . 2  |-  ( ph  ->  G : om --> A )
17 foelrn 5766 . . . . . 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 2248 . . . . . . . . . . 11  |-  ( k  =  x  ->  (
k  e.  N  <->  x  e.  N ) )
20 fveq2 5527 . . . . . . . . . . 11  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2119, 20ifbieq1d 3568 . . . . . . . . . 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 4617 . . . . . . . . . . 11  |-  ( ( x  e.  N  /\  N  e.  om )  ->  x  e.  om )
2522, 23, 24syl2anc 411 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  x  e.  om )
2622iftrued 3553 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  =  ( F `  x ) )
273ffvelcdmda 5664 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  ( F `  x )  e.  A )
2826, 27eqeltrd 2264 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  e.  A )
2915, 21, 25, 28fvmptd3 5622 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  if ( x  e.  N ,  ( F `
 x ) ,  ( F `  (/) ) ) )
3029, 26eqtrd 2220 . . . . . . . 8  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  ( F `  x ) )
3130eqeq2d 2199 . . . . . . 7  |-  ( (
ph  /\  x  e.  N )  ->  (
y  =  ( G `
 x )  <->  y  =  ( F `  x ) ) )
3231rexbidva 2484 . . . . . 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 4620 . . . . . . 7  |-  om  e.  On
3635onelssi 4441 . . . . . 6  |-  ( N  e.  om  ->  N  C_ 
om )
37 ssrexv 3232 . . . . . 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 2560 . 2  |-  ( ph  ->  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) )
42 dffo3 5676 . 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 835    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466    C_ wss 3141   (/)c0 3434   ifcif 3546    |-> cmpt 4076   omcom 4601   -->wf 5224   -onto->wfo 5226   ` cfv 5228
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236
This theorem is referenced by:  enumct  7127
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