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Theorem fidcenumlemrk 7015
Description: Lemma for fidcenum 7017. (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 323 . 2 |- ( ph -> ( ph /\ K C_ N ) )
4 sseq1 3203 . . . . 5 |- ( w = (/) -> ( w C_ N <-> (/) C_ N ) )
54anbi2d 464 . . . 4 |- ( w = (/) -> ( ( ph /\ w C_ N ) <-> ( ph /\ (/) C_ N ) ) )
6 imaeq2 5002 . . . . . 6 |- ( w = (/) -> ( F " w ) = ( F " (/) ) )
76eleq2d 2263 . . . . 5 |- ( w = (/) -> ( X e. ( F " w ) <-> X e. ( F " (/) ) ) )
87notbid 668 . . . . 5 |- ( w = (/) -> ( -. X e. ( F " w ) <-> -. X e. ( F " (/) ) ) )
97, 8orbi12d 794 . . . 4 |- ( w = (/) -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " (/) ) \/ -. X e. ( F " (/) ) ) ) )
105, 9imbi12d 234 . . 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 3203 . . . . 5 |- ( w = j -> ( w C_ N <-> j C_ N ) )
1211anbi2d 464 . . . 4 |- ( w = j -> ( ( ph /\ w C_ N ) <-> ( ph /\ j C_ N ) ) )
13 imaeq2 5002 . . . . . 6 |- ( w = j -> ( F " w ) = ( F " j ) )
1413eleq2d 2263 . . . . 5 |- ( w = j -> ( X e. ( F " w ) <-> X e. ( F " j ) ) )
1514notbid 668 . . . . 5 |- ( w = j -> ( -. X e. ( F " w ) <-> -. X e. ( F " j ) ) )
1614, 15orbi12d 794 . . . 4 |- ( w = j -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) )
1712, 16imbi12d 234 . . 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 3203 . . . . 5 |- ( w = suc j -> ( w C_ N <-> suc j C_ N ) )
1918anbi2d 464 . . . 4 |- ( w = suc j -> ( ( ph /\ w C_ N ) <-> ( ph /\ suc j C_ N ) ) )
20 imaeq2 5002 . . . . . 6 |- ( w = suc j -> ( F " w ) = ( F " suc j ) )
2120eleq2d 2263 . . . . 5 |- ( w = suc j -> ( X e. ( F " w ) <-> X e. ( F " suc j ) ) )
2221notbid 668 . . . . 5 |- ( w = suc j -> ( -. X e. ( F " w ) <-> -. X e. ( F " suc j ) ) )
2321, 22orbi12d 794 . . . 4 |- ( w = suc j -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " suc j ) \/ -. X e. ( F " suc j ) ) ) )
2419, 23imbi12d 234 . . 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 3203 . . . . 5 |- ( w = K -> ( w C_ N <-> K C_ N ) )
2625anbi2d 464 . . . 4 |- ( w = K -> ( ( ph /\ w C_ N ) <-> ( ph /\ K C_ N ) ) )
27 imaeq2 5002 . . . . . 6 |- ( w = K -> ( F " w ) = ( F " K ) )
2827eleq2d 2263 . . . . 5 |- ( w = K -> ( X e. ( F " w ) <-> X e. ( F " K ) ) )
2928notbid 668 . . . . 5 |- ( w = K -> ( -. X e. ( F " w ) <-> -. X e. ( F " K ) ) )
3028, 29orbi12d 794 . . . 4 |- ( w = K -> ( ( X e. ( F " w ) \/ -. X e. ( F " w ) ) <-> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) ) )
3126, 30imbi12d 234 . . 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 3451 . . . . . 6 |- -. X e. (/)
33 ima0 5025 . . . . . . 7 |- ( F " (/) ) = (/)
3433eleq2i 2260 . . . . . 6 |- ( X e. ( F " (/) ) <-> X e. (/) )
3532, 34mtbir 672 . . . . 5 |- -. X e. ( F " (/) )
3635a1i 9 . . . 4 |- ( ( ph /\ (/) C_ N ) -> -. X e. ( F " (/) ) )
3736olcd 735 . . 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 490 . . . . 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 490 . . . . 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 527 . . . . 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 531 . . . . 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 529 . . . . . 6 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ph )
45 sssucid 4447 . . . . . . 7 |- j C_ suc j
4645, 43sstrid 3191 . . . . . 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 528 . . . . . 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 433 . . . . 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 490 . . . . 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 7014 . . . 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 364 . . 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 4633 . 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 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   (/)c0 3447   suc csuc 4397   omcom 4623   "cima 4663   -onto->wfo 5253
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
This theorem is referenced by:  fidcenumlemr  7016  ennnfonelemdc  12559
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