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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  –onto→wfo 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
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