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Theorem exmidfodomrlemreseldju 6816
Description: Lemma for exmidfodomrlemrALT 6819. A variant of eldju 6749. (Contributed by Jim Kingdon, 9-Jul-2022.)
Hypotheses
Ref Expression
exmidfodomrlemreseldju.a  |-  ( ph  ->  A  C_  1o )
exmidfodomrlemreseldju.el  |-  ( ph  ->  B  e.  ( A 1o ) )
Assertion
Ref Expression
exmidfodomrlemreseldju  |-  ( ph  ->  ( ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `
 (/) ) ) )

Proof of Theorem exmidfodomrlemreseldju
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 exmidfodomrlemreseldju.a . . . . . . . . . . 11  |-  ( ph  ->  A  C_  1o )
21sselda 3025 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  1o )
3 el1o 6193 . . . . . . . . . 10  |-  ( x  e.  1o  <->  x  =  (/) )
42, 3sylib 120 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  =  (/) )
54fveq2d 5303 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
(inl  |`  A ) `  x )  =  ( (inl  |`  A ) `  (/) ) )
65eqeq2d 2099 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  <->  B  =  (
(inl  |`  A ) `  (/) ) ) )
7 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
84, 7eqeltrrd 2165 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (/)  e.  A
)
98biantrurd 299 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  (/) )  <->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
106, 9bitrd 186 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  <->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
1110biimpd 142 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  ->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
1211rexlimdva 2489 . . . 4  |-  ( ph  ->  ( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  ->  ( (/) 
e.  A  /\  B  =  ( (inl  |`  A ) `
 (/) ) ) ) )
1312imp 122 . . 3  |-  ( (
ph  /\  E. x  e.  A  B  =  ( (inl  |`  A ) `
 x ) )  ->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) )
1413orcd 687 . 2  |-  ( (
ph  /\  E. x  e.  A  B  =  ( (inl  |`  A ) `
 x ) )  ->  ( ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `
 (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
15 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  1o )  ->  x  e.  1o )
1615, 3sylib 120 . . . . . . . 8  |-  ( (
ph  /\  x  e.  1o )  ->  x  =  (/) )
1716fveq2d 5303 . . . . . . 7  |-  ( (
ph  /\  x  e.  1o )  ->  ( (inr  |`  1o ) `  x
)  =  ( (inr  |`  1o ) `  (/) ) )
1817eqeq2d 2099 . . . . . 6  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  ( (inr  |`  1o ) `
 x )  <->  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
1918biimpd 142 . . . . 5  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  ( (inr  |`  1o ) `
 x )  ->  B  =  ( (inr  |`  1o ) `  (/) ) ) )
2019rexlimdva 2489 . . . 4  |-  ( ph  ->  ( E. x  e.  1o  B  =  ( (inr  |`  1o ) `  x )  ->  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
2120imp 122 . . 3  |-  ( (
ph  /\  E. x  e.  1o  B  =  ( (inr  |`  1o ) `  x ) )  ->  B  =  ( (inr  |`  1o ) `  (/) ) )
2221olcd 688 . 2  |-  ( (
ph  /\  E. x  e.  1o  B  =  ( (inr  |`  1o ) `  x ) )  -> 
( ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
23 exmidfodomrlemreseldju.el . . 3  |-  ( ph  ->  B  e.  ( A 1o ) )
24 eldju 6749 . . 3  |-  ( B  e.  ( A 1o )  <-> 
( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  1o  B  =  ( (inr  |`  1o ) `
 x ) ) )
2523, 24sylib 120 . 2  |-  ( ph  ->  ( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  1o  B  =  ( (inr  |`  1o ) `
 x ) ) )
2614, 22, 25mpjaodan 747 1  |-  ( ph  ->  ( ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   (/)c0 3286    |` cres 4438   ` cfv 5010   1oc1o 6166   ⊔ cdju 6720  inlcinl 6727  inrcinr 6728
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 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  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-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-1st 5903  df-2nd 5904  df-1o 6173  df-dju 6721  df-inl 6729  df-inr 6730
This theorem is referenced by:  exmidfodomrlemrALT  6819
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