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Theorem exmidfodomrlemreseldju 7020
Description: Lemma for exmidfodomrlemrALT 7023. A variant of eldju 6919. (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 3065 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  1o )
3 el1o 6300 . . . . . . . . . 10  |-  ( x  e.  1o  <->  x  =  (/) )
42, 3sylib 121 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  =  (/) )
54fveq2d 5391 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
(inl  |`  A ) `  x )  =  ( (inl  |`  A ) `  (/) ) )
65eqeq2d 2127 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  <->  B  =  (
(inl  |`  A ) `  (/) ) ) )
7 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
84, 7eqeltrrd 2193 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (/)  e.  A
)
98biantrurd 301 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  (/) )  <->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
106, 9bitrd 187 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  <->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
1110biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( B  =  ( (inl  |`  A ) `  x
)  ->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) ) )
1211rexlimdva 2524 . . . 4  |-  ( ph  ->  ( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  ->  ( (/) 
e.  A  /\  B  =  ( (inl  |`  A ) `
 (/) ) ) ) )
1312imp 123 . . 3  |-  ( (
ph  /\  E. x  e.  A  B  =  ( (inl  |`  A ) `
 x ) )  ->  ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) ) )
1413orcd 705 . 2  |-  ( (
ph  /\  E. x  e.  A  B  =  ( (inl  |`  A ) `
 x ) )  ->  ( ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `
 (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
15 simpr 109 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  1o )  ->  x  e.  1o )
1615, 3sylib 121 . . . . . . . 8  |-  ( (
ph  /\  x  e.  1o )  ->  x  =  (/) )
1716fveq2d 5391 . . . . . . 7  |-  ( (
ph  /\  x  e.  1o )  ->  ( (inr  |`  1o ) `  x
)  =  ( (inr  |`  1o ) `  (/) ) )
1817eqeq2d 2127 . . . . . 6  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  ( (inr  |`  1o ) `
 x )  <->  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
1918biimpd 143 . . . . 5  |-  ( (
ph  /\  x  e.  1o )  ->  ( B  =  ( (inr  |`  1o ) `
 x )  ->  B  =  ( (inr  |`  1o ) `  (/) ) ) )
2019rexlimdva 2524 . . . 4  |-  ( ph  ->  ( E. x  e.  1o  B  =  ( (inr  |`  1o ) `  x )  ->  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
2120imp 123 . . 3  |-  ( (
ph  /\  E. x  e.  1o  B  =  ( (inr  |`  1o ) `  x ) )  ->  B  =  ( (inr  |`  1o ) `  (/) ) )
2221olcd 706 . 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 6919 . . 3  |-  ( B  e.  ( A 1o )  <-> 
( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  1o  B  =  ( (inr  |`  1o ) `
 x ) ) )
2523, 24sylib 121 . 2  |-  ( ph  ->  ( E. x  e.  A  B  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  1o  B  =  ( (inr  |`  1o ) `
 x ) ) )
2614, 22, 25mpjaodan 770 1  |-  ( ph  ->  ( ( (/)  e.  A  /\  B  =  (
(inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `
 (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463   E.wrex 2392    C_ wss 3039   (/)c0 3331    |` cres 4509   ` cfv 5091   1oc1o 6272   ⊔ cdju 6888  inlcinl 6896  inrcinr 6897
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 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 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899
This theorem is referenced by:  exmidfodomrlemrALT  7023
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