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Theorem fodjum 6986
Description: Lemma for fodjuomni 6989 and fodjumkv 7002. A condition which shows that  A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjuf.fo  |-  ( ph  ->  F : O -onto-> ( A B ) )
fodjuf.p  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
fodjum.z  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
Assertion
Ref Expression
fodjum  |-  ( ph  ->  E. x  x  e.  A )
Distinct variable groups:    ph, y, z   
y, O, z    z, A    z, B    z, F    w, A, x, z    y, A, w    y, F    ph, w
Allowed substitution hints:    ph( x)    B( x, y, w)    P( x, y, z, w)    F( x, w)    O( x, w)

Proof of Theorem fodjum
StepHypRef Expression
1 fodjum.z . 2  |-  ( ph  ->  E. w  e.  O  ( P `  w )  =  (/) )
2 1n0 6297 . . . . . . . . 9  |-  1o  =/=  (/)
32nesymi 2331 . . . . . . . 8  |-  -.  (/)  =  1o
43intnan 899 . . . . . . 7  |-  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o )
54a1i 9 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  -.  ( -.  E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  1o ) )
6 simprr 506 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  (/) )
7 fodjuf.p . . . . . . . . 9  |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl
`  z ) ,  (/) ,  1o ) )
8 fveqeq2 5398 . . . . . . . . . . 11  |-  ( y  =  w  ->  (
( F `  y
)  =  (inl `  z )  <->  ( F `  w )  =  (inl
`  z ) ) )
98rexbidv 2415 . . . . . . . . . 10  |-  ( y  =  w  ->  ( E. z  e.  A  ( F `  y )  =  (inl `  z
)  <->  E. z  e.  A  ( F `  w )  =  (inl `  z
) ) )
109ifbid 3463 . . . . . . . . 9  |-  ( y  =  w  ->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z
) ,  (/) ,  1o )  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
11 simprl 505 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  w  e.  O )
12 peano1 4478 . . . . . . . . . . 11  |-  (/)  e.  om
1312a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/) 
e.  om )
14 1onn 6384 . . . . . . . . . . 11  |-  1o  e.  om
1514a1i 9 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  1o  e.  om )
16 fodjuf.fo . . . . . . . . . . . 12  |-  ( ph  ->  F : O -onto-> ( A B ) )
1716fodjuomnilemdc 6984 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  O )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
1817adantrr 470 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z ) )
1913, 15, 18ifcldcd 3477 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  if ( E. z  e.  A  ( F `  w )  =  (inl
`  z ) ,  (/) ,  1o )  e. 
om )
207, 10, 11, 19fvmptd3 5482 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( P `  w
)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
216, 20eqtr3d 2152 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) )
22 eqifdc 3476 . . . . . . . 8  |-  (DECID  E. z  e.  A  ( F `  w )  =  (inl
`  z )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2318, 22syl 14 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( (/)  =  if ( E. z  e.  A  ( F `  w )  =  (inl `  z
) ,  (/) ,  1o ) 
<->  ( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) ) )
2421, 23mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( ( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) )  \/  ( -.  E. z  e.  A  ( F `  w )  =  (inl `  z
)  /\  (/)  =  1o ) ) )
255, 24ecased 1312 . . . . 5  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  -> 
( E. z  e.  A  ( F `  w )  =  (inl
`  z )  /\  (/)  =  (/) ) )
2625simpld 111 . . . 4  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  e.  A  ( F `  w )  =  (inl `  z
) )
27 rexm 3432 . . . 4  |-  ( E. z  e.  A  ( F `  w )  =  (inl `  z
)  ->  E. z 
z  e.  A )
2826, 27syl 14 . . 3  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. z  z  e.  A )
29 eleq1w 2178 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
3029cbvexv 1872 . . 3  |-  ( E. z  z  e.  A  <->  E. x  x  e.  A
)
3128, 30sylib 121 . 2  |-  ( (
ph  /\  ( w  e.  O  /\  ( P `  w )  =  (/) ) )  ->  E. x  x  e.  A )
321, 31rexlimddv 2531 1  |-  ( ph  ->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682  DECID wdc 804    = wceq 1316   E.wex 1453    e. wcel 1465   E.wrex 2394   (/)c0 3333   ifcif 3444    |-> cmpt 3959   omcom 4474   -onto->wfo 5091   ` cfv 5093   1oc1o 6274   ⊔ cdju 6890  inlcinl 6898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  fodjuomnilemres  6988  fodjumkvlemres  7001
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