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Theorem fidcenumlemrk 7084
Description: Lemma for fidcenum 7086. (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 3225 . . . . 5 |- ( w = (/) -> ( w C_ N <-> (/) C_ N ) )
54anbi2d 464 . . . 4 |- ( w = (/) -> ( ( ph /\ w C_ N ) <-> ( ph /\ (/) C_ N ) ) )
6 imaeq2 5038 . . . . . 6 |- ( w = (/) -> ( F " w ) = ( F " (/) ) )
76eleq2d 2277 . . . . 5 |- ( w = (/) -> ( X e. ( F " w ) <-> X e. ( F " (/) ) ) )
87notbid 669 . . . . 5 |- ( w = (/) -> ( -. X e. ( F " w ) <-> -. X e. ( F " (/) ) ) )
97, 8orbi12d 795 . . . 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 3225 . . . . 5 |- ( w = j -> ( w C_ N <-> j C_ N ) )
1211anbi2d 464 . . . 4 |- ( w = j -> ( ( ph /\ w C_ N ) <-> ( ph /\ j C_ N ) ) )
13 imaeq2 5038 . . . . . 6 |- ( w = j -> ( F " w ) = ( F " j ) )
1413eleq2d 2277 . . . . 5 |- ( w = j -> ( X e. ( F " w ) <-> X e. ( F " j ) ) )
1514notbid 669 . . . . 5 |- ( w = j -> ( -. X e. ( F " w ) <-> -. X e. ( F " j ) ) )
1614, 15orbi12d 795 . . . 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 3225 . . . . 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 5038 . . . . . 6 |- ( w = suc j -> ( F " w ) = ( F " suc j ) )
2120eleq2d 2277 . . . . 5 |- ( w = suc j -> ( X e. ( F " w ) <-> X e. ( F " suc j ) ) )
2221notbid 669 . . . . 5 |- ( w = suc j -> ( -. X e. ( F " w ) <-> -. X e. ( F " suc j ) ) )
2321, 22orbi12d 795 . . . 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 3225 . . . . 5 |- ( w = K -> ( w C_ N <-> K C_ N ) )
2625anbi2d 464 . . . 4 |- ( w = K -> ( ( ph /\ w C_ N ) <-> ( ph /\ K C_ N ) ) )
27 imaeq2 5038 . . . . . 6 |- ( w = K -> ( F " w ) = ( F " K ) )
2827eleq2d 2277 . . . . 5 |- ( w = K -> ( X e. ( F " w ) <-> X e. ( F " K ) ) )
2928notbid 669 . . . . 5 |- ( w = K -> ( -. X e. ( F " w ) <-> -. X e. ( F " K ) ) )
3028, 29orbi12d 795 . . . 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 3473 . . . . . 6 |- -. X e. (/)
33 ima0 5061 . . . . . . 7 |- ( F " (/) ) = (/)
3433eleq2i 2274 . . . . . 6 |- ( X e. ( F " (/) ) <-> X e. (/) )
3532, 34mtbir 673 . . . . 5 |- -. X e. ( F " (/) )
3635a1i 9 . . . 4 |- ( ( ph /\ (/) C_ N ) -> -. X e. ( F " (/) ) )
3736olcd 736 . . 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 4481 . . . . . . 7 |- j C_ suc j
4645, 43sstrid 3213 . . . . . 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 7083 . . . 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 4667 . 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 710  DECID wdc 836   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3175   (/)c0 3469   suc csuc 4431   omcom 4657   "cima 4697   -onto->wfo 5289
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-iinf 4655
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-sbc 3007  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-id 4359  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-fo 5297  df-fv 5299
This theorem is referenced by:  fidcenumlemr  7085  ennnfonelemdc  12931
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