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Theorem fidcenumlemrk 6987
Description: Lemma for fidcenum 6989. (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 3193 . . . . 5 |- ( w = (/) -> ( w C_ N <-> (/) C_ N ) )
54anbi2d 464 . . . 4 |- ( w = (/) -> ( ( ph /\ w C_ N ) <-> ( ph /\ (/) C_ N ) ) )
6 imaeq2 4987 . . . . . 6 |- ( w = (/) -> ( F " w ) = ( F " (/) ) )
76eleq2d 2259 . . . . 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 3193 . . . . 5 |- ( w = j -> ( w C_ N <-> j C_ N ) )
1211anbi2d 464 . . . 4 |- ( w = j -> ( ( ph /\ w C_ N ) <-> ( ph /\ j C_ N ) ) )
13 imaeq2 4987 . . . . . 6 |- ( w = j -> ( F " w ) = ( F " j ) )
1413eleq2d 2259 . . . . 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 3193 . . . . 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 4987 . . . . . 6 |- ( w = suc j -> ( F " w ) = ( F " suc j ) )
2120eleq2d 2259 . . . . 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 3193 . . . . 5 |- ( w = K -> ( w C_ N <-> K C_ N ) )
2625anbi2d 464 . . . 4 |- ( w = K -> ( ( ph /\ w C_ N ) <-> ( ph /\ K C_ N ) ) )
27 imaeq2 4987 . . . . . 6 |- ( w = K -> ( F " w ) = ( F " K ) )
2827eleq2d 2259 . . . . 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 3441 . . . . . 6 |- -. X e. (/)
33 ima0 5008 . . . . . . 7 |- ( F " (/) ) = (/)
3433eleq2i 2256 . . . . . 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 4436 . . . . . . 7 |- j C_ suc j
4645, 43sstrid 3181 . . . . . 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 6986 . . . 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 4620 . 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 2160   A.wral 2468   C_ wss 3144   (/)c0 3437   suc csuc 4386   omcom 4610   "cima 4650   -onto->wfo 5236
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-iinf 4608
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-id 4314  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fo 5244  df-fv 5246
This theorem is referenced by:  fidcenumlemr  6988  ennnfonelemdc  12461
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