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Theorem enumctlemm 7091
Description: Lemma for enumct 7092. 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 5420 . . . . . . 7  |-  ( F : N -onto-> A  ->  F : N --> A )
31, 2syl 14 . . . . . 6  |-  ( ph  ->  F : N --> A )
43ffvelrnda 5631 . . . . 5  |-  ( (
ph  /\  k  e.  N )  ->  ( F `  k )  e.  A )
54adantlr 474 . . . 4  |-  ( ( ( ph  /\  k  e.  om )  /\  k  e.  N )  ->  ( F `  k )  e.  A )
6 enumctlemm.n0 . . . . . 6  |-  ( ph  -> 
(/)  e.  N )
73, 6ffvelrnd 5632 . . . . 5  |-  ( ph  ->  ( F `  (/) )  e.  A )
87ad2antrr 485 . . . 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 6479 . . . . 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 3555 . . 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 5650 . 2  |-  ( ph  ->  G : om --> A )
17 foelrn 5732 . . . . . 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 2231 . . . . . . . . . . 11  |-  ( k  =  x  ->  (
k  e.  N  <->  x  e.  N ) )
20 fveq2 5496 . . . . . . . . . . 11  |-  ( k  =  x  ->  ( F `  k )  =  ( F `  x ) )
2119, 20ifbieq1d 3548 . . . . . . . . . 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 4590 . . . . . . . . . . 11  |-  ( ( x  e.  N  /\  N  e.  om )  ->  x  e.  om )
2522, 23, 24syl2anc 409 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  x  e.  om )
2622iftrued 3533 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  =  ( F `  x ) )
273ffvelrnda 5631 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  N )  ->  ( F `  x )  e.  A )
2826, 27eqeltrd 2247 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  N )  ->  if ( x  e.  N ,  ( F `  x ) ,  ( F `  (/) ) )  e.  A )
2915, 21, 25, 28fvmptd3 5589 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  if ( x  e.  N ,  ( F `
 x ) ,  ( F `  (/) ) ) )
3029, 26eqtrd 2203 . . . . . . . 8  |-  ( (
ph  /\  x  e.  N )  ->  ( G `  x )  =  ( F `  x ) )
3130eqeq2d 2182 . . . . . . 7  |-  ( (
ph  /\  x  e.  N )  ->  (
y  =  ( G `
 x )  <->  y  =  ( F `  x ) ) )
3231rexbidva 2467 . . . . . 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 4593 . . . . . . 7  |-  om  e.  On
3635onelssi 4414 . . . . . 6  |-  ( N  e.  om  ->  N  C_ 
om )
37 ssrexv 3212 . . . . . 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 2543 . 2  |-  ( ph  ->  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) )
42 dffo3 5643 . 2  |-  ( G : om -onto-> A  <->  ( G : om --> A  /\  A. y  e.  A  E. x  e.  om  y  =  ( G `  x ) ) )
4316, 41, 42sylanbrc 415 1  |-  ( ph  ->  G : om -onto-> A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 829    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   (/)c0 3414   ifcif 3526    |-> cmpt 4050   omcom 4574   -->wf 5194   -onto->wfo 5196   ` cfv 5198
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206
This theorem is referenced by:  enumct  7092
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