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Theorem fodjuomnilemres 6782
Description: Lemma for fodjuomni 6783. The final result with  P broken out into a hypothesis. (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 5288 . . . . . 6  |-  ( f  =  P  ->  (
f `  w )  =  ( P `  w ) )
21eqeq1d 2096 . . . . 5  |-  ( f  =  P  ->  (
( f `  w
)  =  (/)  <->  ( P `  w )  =  (/) ) )
32rexbidv 2381 . . . 4  |-  ( f  =  P  ->  ( E. w  e.  O  ( f `  w
)  =  (/)  <->  E. w  e.  O  ( P `  w )  =  (/) ) )
41eqeq1d 2096 . . . . 5  |-  ( f  =  P  ->  (
( f `  w
)  =  1o  <->  ( P `  w )  =  1o ) )
54ralbidv 2380 . . . 4  |-  ( f  =  P  ->  ( A. w  e.  O  ( f `  w
)  =  1o  <->  A. w  e.  O  ( P `  w )  =  1o ) )
63, 5orbi12d 742 . . 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 6772 . . . . 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 145 . . 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 ) )
137, 11, 12fodjuomnilemf 6779 . . 3  |-  ( ph  ->  P  e.  ( 2o 
^m  O ) )
146, 10, 13rspcdva 2727 . 2  |-  ( ph  ->  ( E. w  e.  O  ( P `  w )  =  (/)  \/ 
A. w  e.  O  ( P `  w )  =  1o ) )
157adantr 270 . . . . 5  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  O  e. Omni )
1611adantr 270 . . . . 5  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  F : O -onto->
( A B )
)
17 simpr 108 . . . . . 6  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. w  e.  O  ( P `  w )  =  (/) )
18 fveq2 5289 . . . . . . . 8  |-  ( w  =  v  ->  ( P `  w )  =  ( P `  v ) )
1918eqeq1d 2096 . . . . . . 7  |-  ( w  =  v  ->  (
( P `  w
)  =  (/)  <->  ( P `  v )  =  (/) ) )
2019cbvrexv 2591 . . . . . 6  |-  ( E. w  e.  O  ( P `  w )  =  (/)  <->  E. v  e.  O  ( P `  v )  =  (/) )
2117, 20sylib 120 . . . . 5  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. v  e.  O  ( P `  v )  =  (/) )
2215, 16, 12, 21fodjuomnilemm 6780 . . . 4  |-  ( (
ph  /\  E. w  e.  O  ( P `  w )  =  (/) )  ->  E. x  x  e.  A )
2322ex 113 . . 3  |-  ( ph  ->  ( E. w  e.  O  ( P `  w )  =  (/)  ->  E. x  x  e.  A ) )
247adantr 270 . . . . 5  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  O  e. Omni )
2511adantr 270 . . . . 5  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  F : O -onto-> ( A B ) )
26 simpr 108 . . . . 5  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  A. w  e.  O  ( P `  w )  =  1o )
2724, 25, 12, 26fodjuomnilem0 6781 . . . 4  |-  ( (
ph  /\  A. w  e.  O  ( P `  w )  =  1o )  ->  A  =  (/) )
2827ex 113 . . 3  |-  ( ph  ->  ( A. w  e.  O  ( P `  w )  =  1o 
->  A  =  (/) ) )
2923, 28orim12d 735 . 2  |-  ( ph  ->  ( ( E. w  e.  O  ( P `  w )  =  (/)  \/ 
A. w  e.  O  ( P `  w )  =  1o )  -> 
( E. x  x  e.  A  \/  A  =  (/) ) ) )
3014, 29mpd 13 1  |-  ( ph  ->  ( E. x  x  e.  A  \/  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360   (/)c0 3284   ifcif 3389    |-> cmpt 3891   -onto->wfo 5000   ` cfv 5002  (class class class)co 5634   1oc1o 6156   2oc2o 6157    ^m cmap 6385   ⊔ cdju 6709  inlcinl 6716  Omnicomni 6767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-1o 6163  df-2o 6164  df-map 6387  df-dju 6710  df-inl 6718  df-inr 6719  df-omni 6769
This theorem is referenced by:  fodjuomni  6783
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