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Theorem fodjuomnilemdc 7245
Description: Lemma for fodjuomni 7250. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypothesis
Ref Expression
fodjuomnilemdc.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
Assertion
Ref Expression
fodjuomnilemdc  |-  ( (
ph  /\  X  e.  O )  -> DECID  E. z  e.  A  ( F `  X )  =  (inl `  z
) )
Distinct variable groups:    z, A    z, B    z, F    z, O    z, X    ph, z

Proof of Theorem fodjuomnilemdc
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 fodjuomnilemdc.fo . . . . . 6  |-  ( ph  ->  F : O -onto-> ( A B ) )
2 fof 5497 . . . . . 6  |-  ( F : O -onto-> ( A B )  ->  F : O --> ( A B ) )
31, 2syl 14 . . . . 5  |-  ( ph  ->  F : O --> ( A B ) )
43ffvelcdmda 5714 . . . 4  |-  ( (
ph  /\  X  e.  O )  ->  ( F `  X )  e.  ( A B )
)
5 djur 7170 . . . 4  |-  ( ( F `  X )  e.  ( A B )  <-> 
( E. z  e.  A  ( F `  X )  =  (inl
`  z )  \/ 
E. z  e.  B  ( F `  X )  =  (inr `  z
) ) )
64, 5sylib 122 . . 3  |-  ( (
ph  /\  X  e.  O )  ->  ( E. z  e.  A  ( F `  X )  =  (inl `  z
)  \/  E. z  e.  B  ( F `  X )  =  (inr
`  z ) ) )
7 nfv 1550 . . . . . . . 8  |-  F/ z ( ph  /\  X  e.  O )
8 nfre1 2548 . . . . . . . 8  |-  F/ z E. z  e.  B  ( F `  X )  =  (inr `  z
)
97, 8nfan 1587 . . . . . . 7  |-  F/ z ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z
) )
10 simpr 110 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  E. z  e.  B  ( F `  X )  =  (inr
`  z ) )
11 fveq2 5575 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (inr `  z )  =  (inr
`  w ) )
1211eqeq2d 2216 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  X
)  =  (inr `  z )  <->  ( F `  X )  =  (inr
`  w ) ) )
1312cbvrexv 2738 . . . . . . . . . 10  |-  ( E. z  e.  B  ( F `  X )  =  (inr `  z
)  <->  E. w  e.  B  ( F `  X )  =  (inr `  w
) )
1410, 13sylib 122 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  E. w  e.  B  ( F `  X )  =  (inr
`  w ) )
15 vex 2774 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
16 vex 2774 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
17 djune 7179 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  _V  /\  w  e.  _V )  ->  (inl `  z )  =/=  (inr `  w )
)
1815, 16, 17mp2an 426 . . . . . . . . . . . . . 14  |-  (inl `  z )  =/=  (inr `  w )
19 neeq2 2389 . . . . . . . . . . . . . 14  |-  ( ( F `  X )  =  (inr `  w
)  ->  ( (inl `  z )  =/=  ( F `  X )  <->  (inl
`  z )  =/=  (inr `  w )
) )
2018, 19mpbiri 168 . . . . . . . . . . . . 13  |-  ( ( F `  X )  =  (inr `  w
)  ->  (inl `  z
)  =/=  ( F `
 X ) )
2120necomd 2461 . . . . . . . . . . . 12  |-  ( ( F `  X )  =  (inr `  w
)  ->  ( F `  X )  =/=  (inl `  z ) )
2221neneqd 2396 . . . . . . . . . . 11  |-  ( ( F `  X )  =  (inr `  w
)  ->  -.  ( F `  X )  =  (inl `  z )
)
2322a1i 9 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  ( ( F `  X )  =  (inr `  w )  ->  -.  ( F `  X )  =  (inl
`  z ) ) )
2423rexlimdvw 2626 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  ( E. w  e.  B  ( F `  X )  =  (inr `  w )  ->  -.  ( F `  X )  =  (inl
`  z ) ) )
2514, 24mpd 13 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  -.  ( F `  X )  =  (inl `  z )
)
2625a1d 22 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  ( z  e.  A  ->  -.  ( F `  X )  =  (inl `  z )
) )
279, 26ralrimi 2576 . . . . . 6  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  A. z  e.  A  -.  ( F `  X )  =  (inl `  z )
)
28 ralnex 2493 . . . . . 6  |-  ( A. z  e.  A  -.  ( F `  X )  =  (inl `  z
)  <->  -.  E. z  e.  A  ( F `  X )  =  (inl
`  z ) )
2927, 28sylib 122 . . . . 5  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  -.  E. z  e.  A  ( F `  X )  =  (inl
`  z ) )
3029ex 115 . . . 4  |-  ( (
ph  /\  X  e.  O )  ->  ( E. z  e.  B  ( F `  X )  =  (inr `  z
)  ->  -.  E. z  e.  A  ( F `  X )  =  (inl
`  z ) ) )
3130orim2d 789 . . 3  |-  ( (
ph  /\  X  e.  O )  ->  (
( E. z  e.  A  ( F `  X )  =  (inl
`  z )  \/ 
E. z  e.  B  ( F `  X )  =  (inr `  z
) )  ->  ( E. z  e.  A  ( F `  X )  =  (inl `  z
)  \/  -.  E. z  e.  A  ( F `  X )  =  (inl `  z )
) ) )
326, 31mpd 13 . 2  |-  ( (
ph  /\  X  e.  O )  ->  ( E. z  e.  A  ( F `  X )  =  (inl `  z
)  \/  -.  E. z  e.  A  ( F `  X )  =  (inl `  z )
) )
33 df-dc 836 . 2  |-  (DECID  E. z  e.  A  ( F `  X )  =  (inl
`  z )  <->  ( E. z  e.  A  ( F `  X )  =  (inl `  z )  \/  -.  E. z  e.  A  ( F `  X )  =  (inl
`  z ) ) )
3432, 33sylibr 134 1  |-  ( (
ph  /\  X  e.  O )  -> DECID  E. z  e.  A  ( F `  X )  =  (inl `  z
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    = wceq 1372    e. wcel 2175    =/= wne 2375   A.wral 2483   E.wrex 2484   _Vcvv 2771   -->wf 5266   -onto->wfo 5268   ` cfv 5270   ⊔ cdju 7138  inlcinl 7146  inrcinr 7147
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-dju 7139  df-inl 7148  df-inr 7149
This theorem is referenced by:  fodjuf  7246  fodjum  7247  fodju0  7248
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