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Theorem enumctlemm 7058
Description: Lemma for enumct 7059. 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 5392 . . . . . . 7  |-  ( F : N -onto-> A  ->  F : N --> A )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  F : N --> A )
43ffvelrnda 5602 . . . . 5  |-  ( (
ph  /\  k  e.  N )  ->  ( F `  k )  e.  A )
54adantlr 469 . . . 4  |-  ( ( ( ph  /\  k  e.  om )  /\  k  e.  N )  ->  ( F `  k )  e.  A )
6 enumctlemm.n0 . . . . . 6  |-  ( ph  -> 
(/)  e.  N )
73, 6ffvelrnd 5603 . . . . 5  |-  ( ph  ->  ( F `  (/) )  e.  A )
87ad2antrr 480 . . . 4  |-  ( ( ( ph  /\  k  e.  om )  /\  -.  k  e.  N )  ->  ( F `  (/) )  e.  A )
9 simpr 109 . . . . 5  |-  ( (
ph  /\  k  e.  om )  ->  k  e.  om )
10 enumctlemm.n . . . . . 6  |-  ( ph  ->  N  e.  om )
1110adantr 274 . . . . 5  |-  ( (
ph  /\  k  e.  om )  ->  N  e.  om )
12 nndcel 6447 . . . . 5  |-  ( ( k  e.  om  /\  N  e.  om )  -> DECID  k  e.  N )
139, 11, 12syl2anc 409 . . . 4  |-  ( (
ph  /\  k  e.  om )  -> DECID  k  e.  N
)
145, 8, 13ifcldadc 3534 . . 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 5621 . 2  |-  ( ph  ->  G : om --> A )
17 foelrn 5703 . . . . . 6  |-  ( ( F : N -onto-> A  /\  y  e.  A
)  ->  E. x  e.  N  y  =  ( F `  x ) )
181, 17sylan 281 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  E. x  e.  N  y  =  ( F `  x ) )
19 eleq1w 2218 . . . . . . . . . . 11  |-  ( k  =  x  ->  (
k  e.  N  <->  x  e.  N ) )
20 fveq2 5468 . . . . . . . . . . 11  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2119, 20ifbieq1d 3527 . . . . . . . . . 10  |-  ( k  =  x  ->  if ( k  e.  N ,  ( F `  k ) ,  ( F `  (/) ) )  =  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) ) )
22 simpr 109 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  x  e.  N )
2310adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  N  e.  om )
24 elnn 4565 . . . . . . . . . . 11  |-  ( ( x  e.  N  /\  N  e.  om )  ->  x  e.  om )
2522, 23, 24syl2anc 409 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  x  e.  om )
2622iftrued 3512 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  =  ( F `  x ) )
273ffvelrnda 5602 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  ( F `  x )  e.  A )
2826, 27eqeltrd 2234 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  e.  A )
2915, 21, 25, 28fvmptd3 5561 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  if ( x  e.  N ,  ( F `
 x ) ,  ( F `  (/) ) ) )
3029, 26eqtrd 2190 . . . . . . . 8  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  ( F `  x ) )
3130eqeq2d 2169 . . . . . . 7  |-  ( (
ph  /\  x  e.  N )  ->  (
y  =  ( G `
 x )  <->  y  =  ( F `  x ) ) )
3231rexbidva 2454 . . . . . 6  |-  ( ph  ->  ( E. x  e.  N  y  =  ( G `  x )  <->  E. x  e.  N  y  =  ( F `  x ) ) )
3332adantr 274 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  ( E. x  e.  N  y  =  ( G `  x )  <->  E. x  e.  N  y  =  ( F `  x ) ) )
3418, 33mpbird 166 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  E. x  e.  N  y  =  ( G `  x ) )
35 omelon 4568 . . . . . . 7  |-  om  e.  On
3635onelssi 4389 . . . . . 6  |-  ( N  e.  om  ->  N  C_ 
om )
37 ssrexv 3193 . . . . . 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 274 . . . 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 2530 . 2  |-  ( ph  ->  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) )
42 dffo3 5614 . 2  |-  ( G : om -onto-> A  <->  ( G : om --> A  /\  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) ) )
4316, 41, 42sylanbrc 414 1  |-  ( ph  ->  G : om -onto-> A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 820    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436    C_ wss 3102   (/)c0 3394   ifcif 3505    |-> cmpt 4025   omcom 4549   -->wf 5166   -onto->wfo 5168   ` cfv 5170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176  df-fv 5178
This theorem is referenced by:  enumct  7059
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