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Theorem fodjuomnilemdc 7136
Description: Lemma for fodjuomni 7141. 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 5434 . . . . . 6  |-  ( F : O -onto-> ( A B )  ->  F : O --> ( A B ) )
31, 2syl 14 . . . . 5  |-  ( ph  ->  F : O --> ( A B ) )
43ffvelcdmda 5647 . . . 4  |-  ( (
ph  /\  X  e.  O )  ->  ( F `  X )  e.  ( A B )
)
5 djur 7062 . . . 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 1528 . . . . . . . 8  |-  F/ z ( ph  /\  X  e.  O )
8 nfre1 2520 . . . . . . . 8  |-  F/ z E. z  e.  B  ( F `  X )  =  (inr `  z
)
97, 8nfan 1565 . . . . . . 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 5511 . . . . . . . . . . . 12  |-  ( z  =  w  ->  (inr `  z )  =  (inr
`  w ) )
1211eqeq2d 2189 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  X
)  =  (inr `  z )  <->  ( F `  X )  =  (inr
`  w ) ) )
1312cbvrexv 2704 . . . . . . . . . 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 2740 . . . . . . . . . . . . . . 15  |-  z  e. 
_V
16 vex 2740 . . . . . . . . . . . . . . 15  |-  w  e. 
_V
17 djune 7071 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  _V  /\  w  e.  _V )  ->  (inl `  z )  =/=  (inr `  w )
)
1815, 16, 17mp2an 426 . . . . . . . . . . . . . 14  |-  (inl `  z )  =/=  (inr `  w )
19 neeq2 2361 . . . . . . . . . . . . . 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 2433 . . . . . . . . . . . 12  |-  ( ( F `  X )  =  (inr `  w
)  ->  ( F `  X )  =/=  (inl `  z ) )
2221neneqd 2368 . . . . . . . . . . 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 2598 . . . . . . . . 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 2548 . . . . . 6  |-  ( ( ( ph  /\  X  e.  O )  /\  E. z  e.  B  ( F `  X )  =  (inr `  z )
)  ->  A. z  e.  A  -.  ( F `  X )  =  (inl `  z )
)
28 ralnex 2465 . . . . . 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 788 . . 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 835 . 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 708  DECID wdc 834    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455   E.wrex 2456   _Vcvv 2737   -->wf 5208   -onto->wfo 5210   ` cfv 5212   ⊔ cdju 7030  inlcinl 7038  inrcinr 7039
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041
This theorem is referenced by:  fodjuf  7137  fodjum  7138  fodju0  7139
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