ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fodjuomni GIF version

Theorem fodjuomni 7025
Description: A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o (𝜑𝑂 ∈ Omni)
fodjuomni.fo (𝜑𝐹:𝑂onto→(𝐴𝐵))
Assertion
Ref Expression
fodjuomni (𝜑 → (∃𝑥 𝑥𝐴𝐴 = ∅))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem fodjuomni
Dummy variables 𝑎 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuomni.o . 2 (𝜑𝑂 ∈ Omni)
2 fodjuomni.fo . 2 (𝜑𝐹:𝑂onto→(𝐴𝐵))
3 fveq2 5425 . . . . . . 7 (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧))
43eqeq2d 2152 . . . . . 6 (𝑏 = 𝑧 → ((𝐹𝑎) = (inl‘𝑏) ↔ (𝐹𝑎) = (inl‘𝑧)))
54cbvrexv 2656 . . . . 5 (∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧))
6 ifbi 3493 . . . . 5 ((∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏) ↔ ∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧)) → if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
75, 6ax-mp 5 . . . 4 if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)
87mpteq2i 4019 . . 3 (𝑎𝑂 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o))
9 fveq2 5425 . . . . . . 7 (𝑎 = 𝑦 → (𝐹𝑎) = (𝐹𝑦))
109eqeq1d 2149 . . . . . 6 (𝑎 = 𝑦 → ((𝐹𝑎) = (inl‘𝑧) ↔ (𝐹𝑦) = (inl‘𝑧)))
1110rexbidv 2439 . . . . 5 (𝑎 = 𝑦 → (∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧)))
1211ifbid 3494 . . . 4 (𝑎 = 𝑦 → if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
1312cbvmptv 4028 . . 3 (𝑎𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
148, 13eqtri 2161 . 2 (𝑎𝑂 ↦ if(∃𝑏𝐴 (𝐹𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
151, 2, 14fodjuomnilemres 7024 1 (𝜑 → (∃𝑥 𝑥𝐴𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 698   = wceq 1332  wex 1469  wcel 1481  wrex 2418  c0 3364  ifcif 3475  cmpt 3993  ontowfo 5125  cfv 5127  1oc1o 6310  cdju 6926  inlcinl 6934  Omnicomni 7008
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-if 3476  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-1o 6317  df-2o 6318  df-map 6548  df-dju 6927  df-inl 6936  df-inr 6937  df-omni 7010
This theorem is referenced by:  ctssexmid  7028  exmidunben  11966  exmidsbthrlem  13375  sbthomlem  13378
  Copyright terms: Public domain W3C validator