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Theorem fidcenumlemrk 7261
Description: Lemma for fidcenum 7263. (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 3271 . . . . 5 |- ( w = (/) -> ( w C_ N <-> (/) C_ N ) )
54anbi2d 468 . . . 4 |- ( w = (/) -> ( ( ph /\ w C_ N ) <-> ( ph /\ (/) C_ N ) ) )
6 imaeq2 5117 . . . . . 6 |- ( w = (/) -> ( F " w ) = ( F " (/) ) )
76eleq2d 2308 . . . . 5 |- ( w = (/) -> ( X e. ( F " w ) <-> X e. ( F " (/) ) ) )
87notbid 677 . . . . 5 |- ( w = (/) -> ( -. X e. ( F " w ) <-> -. X e. ( F " (/) ) ) )
97, 8orbi12d 805 . . . 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 3271 . . . . 5 |- ( w = j -> ( w C_ N <-> j C_ N ) )
1211anbi2d 468 . . . 4 |- ( w = j -> ( ( ph /\ w C_ N ) <-> ( ph /\ j C_ N ) ) )
13 imaeq2 5117 . . . . . 6 |- ( w = j -> ( F " w ) = ( F " j ) )
1413eleq2d 2308 . . . . 5 |- ( w = j -> ( X e. ( F " w ) <-> X e. ( F " j ) ) )
1514notbid 677 . . . . 5 |- ( w = j -> ( -. X e. ( F " w ) <-> -. X e. ( F " j ) ) )
1614, 15orbi12d 805 . . . 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 3271 . . . . 5 |- ( w = suc j -> ( w C_ N <-> suc j C_ N ) )
1918anbi2d 468 . . . 4 |- ( w = suc j -> ( ( ph /\ w C_ N ) <-> ( ph /\ suc j C_ N ) ) )
20 imaeq2 5117 . . . . . 6 |- ( w = suc j -> ( F " w ) = ( F " suc j ) )
2120eleq2d 2308 . . . . 5 |- ( w = suc j -> ( X e. ( F " w ) <-> X e. ( F " suc j ) ) )
2221notbid 677 . . . . 5 |- ( w = suc j -> ( -. X e. ( F " w ) <-> -. X e. ( F " suc j ) ) )
2321, 22orbi12d 805 . . . 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 3271 . . . . 5 |- ( w = K -> ( w C_ N <-> K C_ N ) )
2625anbi2d 468 . . . 4 |- ( w = K -> ( ( ph /\ w C_ N ) <-> ( ph /\ K C_ N ) ) )
27 imaeq2 5117 . . . . . 6 |- ( w = K -> ( F " w ) = ( F " K ) )
2827eleq2d 2308 . . . . 5 |- ( w = K -> ( X e. ( F " w ) <-> X e. ( F " K ) ) )
2928notbid 677 . . . . 5 |- ( w = K -> ( -. X e. ( F " w ) <-> -. X e. ( F " K ) ) )
3028, 29orbi12d 805 . . . 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 3525 . . . . . 6 |- -. X e. (/)
33 ima0 5141 . . . . . . 7 |- ( F " (/) ) = (/)
3433eleq2i 2305 . . . . . 6 |- ( X e. ( F " (/) ) <-> X e. (/) )
3532, 34mtbir 682 . . . . 5 |- -. X e. ( F " (/) )
3635a1i 9 . . . 4 |- ( ( ph /\ (/) C_ N ) -> -. X e. ( F " (/) ) )
3736olcd 746 . . 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 494 . . . . 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 494 . . . . 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 531 . . . . 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 537 . . . . 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 535 . . . . . 6 |- ( ( ( j e. om /\ ( ( ph /\ j C_ N ) -> ( X e. ( F " j ) \/ -. X e. ( F " j ) ) ) ) /\ ( ph /\ suc j C_ N ) ) -> ph )
45 sssucid 4555 . . . . . . 7 |- j C_ suc j
4645, 43sstrid 3259 . . . . . 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 533 . . . . . 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 437 . . . . 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 494 . . . . 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 7260 . . . 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 4742 . 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 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   (/)c0 3520   suc csuc 4505   omcom 4732   "cima 4772   -onto->wfo 5370
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380
This theorem is referenced by:  fidcenumlemr  7262  ennnfonelemdc  13268
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