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Theorem fidcenumlemrk 6792
Description: Lemma for fidcenum 6794. (Contributed by Jim Kingdon, 20-Oct-2022.)
Hypotheses
Ref Expression
fidcenumlemr.dc |- ( ph -> A. x e. A A. y e. A DECID x = y )
fidcenumlemr.f |- ( ph -> F : N -onto-> A )
fidcenumlemrk.k |- ( ph -> K e. om )
fidcenumlemrk.kn |- ( ph -> K C_ N )
fidcenumlemrk.x |- ( ph -> X e. A )
Assertion
Ref Expression
fidcenumlemrk |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) )
Distinct variable groups:   x, F, y   x, X, y   x, A, y
Allowed substitution hints:   ph( x, y)   K( x, y)   N( x, y)
Proof of Theorem fidcenumlemrk
Dummy variables j w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fidcenumlemrk.k . 2 |- ( ph -> K e. om )
2 fidcenumlemrk.kn . . 3 |- ( ph -> K C_ N )
32ancli 319 . 2 |- ( ph -> ( ph /\ K C_ N ) )
4 sseq1 3084 . . . . 5 |- ( w = (/) -> ( w C_ N <-> (/) C_ N ) )
54anbi2d 457 . . . 4 |- ( w = (/) -> ( ( ph /\ w C_ N ) <-> ( ph /\ (/) C_ N ) ) )
6 imaeq2 4833 . . . . . 6 |- ( w = (/) -> ( F " w ) = ( F " (/) ) )
76eleq2d 2182 . . . . 5 |- ( w = (/) -> ( X e. ( F " w ) <-> X e. ( F " (/) ) ) )
87notbid 639 . . . . 5 |- ( w = (/) -> ( -. X e. ( F " w ) <-> -. X e. ( F " (/) ) ) )
97, 8orbi12d 765 . . . 4 |- ( w = (/) -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " (/) ) \/ -. X e. ( F " (/) ) ) ) )
105, 9imbi12d 233 . . 3 |- ( w = (/) -> ( ( ( ph /\ w C_ N ) -> ( X e. ( F " w ) \/ -. X e. ( F " w ) ) ) <-> ( ( ph /\ (/) C_ N ) -> ( X e. ( F " (/) ) \/ -. X e. ( F " (/) ) ) ) ) )
11 sseq1 3084 . . . . 5 |- ( w = j -> ( w C_ N <-> j C_ N ) )
1211anbi2d 457 . . . 4 |- ( w = j -> ( ( ph /\ w C_ N ) <-> ( ph /\ j C_ N ) ) )
13 imaeq2 4833 . . . . . 6 |- ( w = j -> ( F " w ) = ( F " j ) )
1413eleq2d 2182 . . . . 5 |- ( w = j -> ( X e. ( F " w ) <-> X e. ( F " j ) ) )
1514notbid 639 . . . . 5 |- ( w = j -> ( -. X e. ( F " w ) <-> -. X e. ( F " j ) ) )
1614, 15orbi12d 765 . . . 4 |- ( w = j -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) )
1712, 16imbi12d 233 . . 3 |- ( w = j -> ( ( ( ph /\ w C_ N ) -> ( X e. ( F " w ) \/ -. X e. ( F " w ) ) ) <-> ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) )
18 sseq1 3084 . . . . 5 |- ( w = suc j -> ( w C_ N <-> suc j C_ N ) )
1918anbi2d 457 . . . 4 |- ( w = suc j -> ( ( ph /\ w C_ N ) <-> ( ph /\ suc j C_ N ) ) )
20 imaeq2 4833 . . . . . 6 |- ( w = suc j -> ( F " w ) = ( F " suc j ) )
2120eleq2d 2182 . . . . 5 |- ( w = suc j -> ( X e. ( F " w ) <-> X e. ( F " suc j ) ) )
2221notbid 639 . . . . 5 |- ( w = suc j -> ( -. X e. ( F " w ) <-> -. X e. ( F " suc j ) ) )
2321, 22orbi12d 765 . . . 4 |- ( w = suc j -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " suc j ) \/ -. X e. ( F " suc j ) ) ) )
2419, 23imbi12d 233 . . 3 |- ( w = suc j -> ( ( ( ph /\ w C_ N ) -> ( X e. ( F " w ) \/ -. X e. ( F " w ) ) ) <-> ( ( ph /\ suc j C_ N ) -> ( X e. ( F " suc j ) \/ -. X e. ( F " suc j ) ) ) ) )
25 sseq1 3084 . . . . 5 |- ( w = K -> ( w C_ N <-> K C_ N ) )
2625anbi2d 457 . . . 4 |- ( w = K -> ( ( ph /\ w C_ N ) <-> ( ph /\ K C_ N ) ) )
27 imaeq2 4833 . . . . . 6 |- ( w = K -> ( F " w ) = ( F " K ) )
2827eleq2d 2182 . . . . 5 |- ( w = K -> ( X e. ( F " w ) <-> X e. ( F " K ) ) )
2928notbid 639 . . . . 5 |- ( w = K -> ( -. X e. ( F " w ) <-> -. X e. ( F " K ) ) )
3028, 29orbi12d 765 . . . 4 |- ( w = K -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) ) )
3126, 30imbi12d 233 . . 3 |- ( w = K -> ( ( ( ph /\ w C_ N ) -> ( X e. ( F " w ) \/ -. X e. ( F " w ) ) ) <-> ( ( ph /\ K C_ N ) -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) ) ) )
32 noel 3331 . . . . . 6 |- -. X e. (/)
33 ima0 4854 . . . . . . 7 |- ( F " (/) ) = (/)
3433eleq2i 2179 . . . . . 6 |- ( X e. ( F " (/) ) <-> X e. (/) )
3532, 34mtbir 643 . . . . 5 |- -. X e. ( F " (/) )
3635a1i 9 . . . 4 |- ( ( ph /\ (/) C_ N ) -> -. X e. ( F " (/) ) )
3736olcd 706 . . 3 |- ( ( ph /\ (/) C_ N ) -> ( X e. ( F " (/) ) \/ -. X e. ( F " (/) ) ) )
38 fidcenumlemr.dc . . . . . 6 |- ( ph -> A. x e. A A. y e. A DECID x = y )
3938ad2antrl 479 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> A. x e. A A. y e. A DECID x = y )
40 fidcenumlemr.f . . . . . 6 |- ( ph -> F : N -onto-> A )
4140ad2antrl 479 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> F : N -onto-> A )
42 simpll 501 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> j e. om )
43 simprr 504 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> suc j C_ N )
44 simprl 503 . . . . . 6 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ph )
45 sssucid 4295 . . . . . . 7 |- j C_ suc j
4645, 43syl5ss 3072 . . . . . 6 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> j C_ N )
47 simplr 502 . . . . . 6 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) )
4844, 46, 47mp2and 427 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) )
49 fidcenumlemrk.x . . . . . 6 |- ( ph -> X e. A )
5049ad2antrl 479 . . . . 5 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> X e. A )
5139, 41, 42, 43, 48, 50fidcenumlemrks 6791 . . . 4 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ( X e. ( F " suc j ) \/ -. X e. ( F " suc j ) ) )
5251exp31 359 . . 3 |- ( j e. om -> ( ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) -> ( ( ph /\ suc j C_ N ) -> ( X e. ( F " suc j ) \/ -. X e. ( F " suc j ) ) ) ) )
5310, 17, 24, 31, 37, 52finds 4472 . 2 |- ( K e. om -> ( ( ph /\ K C_ N ) -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) ) )
541, 3, 53sylc 62 1 |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 680  DECID wdc 802   = wceq 1312   e. wcel 1461   A.wral 2388   C_ wss 3035   (/)c0 3327   suc csuc 4245   omcom 4462   "cima 4500   -onto->wfo 5077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-id 4173  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fo 5085  df-fv 5087
This theorem is referenced by:  fidcenumlemr  6793  ennnfonelemdc  11751
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