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Theorem fodjuomnilemres 6970
Description: Lemma for fodjuomni 6971. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o  |-  ( ph  ->  O  e. Omni )
fodjuomni.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
fodjuomni.p  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
Assertion
Ref Expression
fodjuomnilemres  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    x, A, z    y, A   
y, F    y, P, z
Allowed substitution hints:    ph( x)    B( x, y)    P( x)    F( x)    O( x)

Proof of Theorem fodjuomnilemres
Dummy variables  v  f  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5374 . . . . . 6  |-  ( f  =  P  ->  (
f `  w )  =  ( P `  w ) )
21eqeq1d 2123 . . . . 5  |-  ( f  =  P  ->  (
( f `  w
)  =  (/)  <->  ( P `  w )  =  (/) ) )
32rexbidv 2412 . . . 4  |-  ( f  =  P  ->  ( E. w  e.  O  ( f `  w
)  =  (/)  <->  E. w  e.  O  ( P `  w )  =  (/) ) )
41eqeq1d 2123 . . . . 5  |-  ( f  =  P  ->  (
( f `  w
)  =  1o  <->  ( P `  w )  =  1o ) )
54ralbidv 2411 . . . 4  |-  ( f  =  P  ->  ( A. w  e.  O  ( f `  w
)  =  1o  <->  A. w  e.  O  ( P `  w )  =  1o ) )
63, 5orbi12d 765 . . 3  |-  ( f  =  P  ->  (
( E. w  e.  O  ( f `  w )  =  (/)  \/ 
A. w  e.  O  ( f `  w
)  =  1o )  <-> 
( E. w  e.  O  ( P `  w )  =  (/)  \/ 
A. w  e.  O  ( P `  w )  =  1o ) ) )
7 fodjuomni.o . . . 4  |-  ( ph  ->  O  e. Omni )
8 isomnimap 6959 . . . . 5  |-  ( O  e. Omni  ->  ( O  e. Omni  <->  A. f  e.  ( 2o 
^m  O ) ( E. w  e.  O  ( f `  w
)  =  (/)  \/  A. w  e.  O  (
f `  w )  =  1o ) ) )
97, 8syl 14 . . . 4  |-  ( ph  ->  ( O  e. Omni  <->  A. f  e.  ( 2o  ^m  O
) ( E. w  e.  O  ( f `  w )  =  (/)  \/ 
A. w  e.  O  ( f `  w
)  =  1o ) ) )
107, 9mpbid 146 . . 3  |-  ( ph  ->  A. f  e.  ( 2o  ^m  O ) ( E. w  e.  O  ( f `  w )  =  (/)  \/ 
A. w  e.  O  ( f `  w
)  =  1o ) )
11 fodjuomni.fo . . . 4  |-  ( ph  ->  F : O -onto-> ( A B ) )
12 fodjuomni.p . . . 4  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
1311, 12, 7fodjuf 6967 . . 3  |-  ( ph  ->  P  e.  ( 2o 
^m  O ) )
146, 10, 13rspcdva 2765 . 2  |-  ( ph  ->  ( E. w  e.  O  ( P `  w )  =  (/)  \/ 
A. w  e.  O  ( P `  w )  =  1o ) )
1511adantr 272 . . . . 5  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  F : O -onto->
( A B )
)
16 simpr 109 . . . . . 6  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. w  e.  O  ( P `  w )  =  (/) )
17 fveqeq2 5384 . . . . . . 7  |-  ( w  =  v  ->  (
( P `  w
)  =  (/)  <->  ( P `  v )  =  (/) ) )
1817cbvrexv 2629 . . . . . 6  |-  ( E. w  e.  O  ( P `  w )  =  (/)  <->  E. v  e.  O  ( P `  v )  =  (/) )
1916, 18sylib 121 . . . . 5  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. v  e.  O  ( P `  v )  =  (/) )
2015, 12, 19fodjum 6968 . . . 4  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. x  x  e.  A )
2120ex 114 . . 3  |-  ( ph  ->  ( E. w  e.  O  ( P `  w )  =  (/)  ->  E. x  x  e.  A ) )
2211adantr 272 . . . . 5  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  F : O -onto-> ( A B ) )
23 simpr 109 . . . . 5  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  A. w  e.  O  ( P `  w )  =  1o )
2422, 12, 23fodju0 6969 . . . 4  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  A  =  (/) )
2524ex 114 . . 3  |-  ( ph  ->  ( A. w  e.  O  ( P `  w )  =  1o 
->  A  =  (/) ) )
2621, 25orim12d 758 . 2  |-  ( ph  ->  ( ( E. w  e.  O  ( P `  w )  =  (/)  \/ 
A. w  e.  O  ( P `  w )  =  1o )  -> 
( E. x  x  e.  A  \/  A  =  (/) ) ) )
2714, 26mpd 13 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2390   E.wrex 2391   (/)c0 3329   ifcif 3440    |-> cmpt 3949   -onto->wfo 5079   ` cfv 5081  (class class class)co 5728   1oc1o 6260   2oc2o 6261    ^m cmap 6496   ⊔ cdju 6874  inlcinl 6882  Omnicomni 6954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-1o 6267  df-2o 6268  df-map 6498  df-dju 6875  df-inl 6884  df-inr 6885  df-omni 6956
This theorem is referenced by:  fodjuomni  6971
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