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Theorem exmidfodomrlemreseldju 7199
Description: Lemma for exmidfodomrlemrALT 7202. A variant of eldju 7067. (Contributed by Jim Kingdon, 9-Jul-2022.)
Hypotheses
Ref Expression
exmidfodomrlemreseldju.a (𝜑𝐴 ⊆ 1o)
exmidfodomrlemreseldju.el (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
Assertion
Ref Expression
exmidfodomrlemreseldju (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Proof of Theorem exmidfodomrlemreseldju
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 exmidfodomrlemreseldju.a . . . . . . . . . . 11 (𝜑𝐴 ⊆ 1o)
21sselda 3156 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6438 . . . . . . . . . 10 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 122 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5520 . . . . . . . 8 ((𝜑𝑥𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
65eqeq2d 2189 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7 simpr 110 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
84, 7eqeltrrd 2255 . . . . . . . 8 ((𝜑𝑥𝐴) → ∅ ∈ 𝐴)
98biantrurd 305 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘∅) ↔ (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
106, 9bitrd 188 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1110biimpd 144 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1211rexlimdva 2594 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1312imp 124 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)))
1413orcd 733 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
15 simpr 110 . . . . . . . . 9 ((𝜑𝑥 ∈ 1o) → 𝑥 ∈ 1o)
1615, 3sylib 122 . . . . . . . 8 ((𝜑𝑥 ∈ 1o) → 𝑥 = ∅)
1716fveq2d 5520 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
1817eqeq2d 2189 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) ↔ 𝐵 = ((inr ↾ 1o)‘∅)))
1918biimpd 144 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
2019rexlimdva 2594 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
2120imp 124 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → 𝐵 = ((inr ↾ 1o)‘∅))
2221olcd 734 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
23 exmidfodomrlemreseldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
24 eldju 7067 . . 3 (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
2523, 24sylib 122 . 2 (𝜑 → (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
2614, 22, 25mpjaodan 798 1 (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  wrex 2456  wss 3130  c0 3423  cres 4629  cfv 5217  1oc1o 6410  cdju 7036  inlcinl 7044  inrcinr 7045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047
This theorem is referenced by:  exmidfodomrlemrALT  7202
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