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

Theorem fodjuomnilemres 7181
Description: Lemma for fodjuomni 7182. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
Hypotheses
Ref Expression
fodjuomni.o (𝜑𝑂 ∈ Omni)
fodjuomni.fo (𝜑𝐹:𝑂onto→(𝐴𝐵))
fodjuomni.p 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
Assertion
Ref Expression
fodjuomnilemres (𝜑 → (∃𝑥 𝑥𝐴𝐴 = ∅))
Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑥,𝐴,𝑧   𝑦,𝐴   𝑦,𝐹   𝑦,𝑃,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑦)   𝑃(𝑥)   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem fodjuomnilemres
Dummy variables 𝑣 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5536 . . . . . 6 (𝑓 = 𝑃 → (𝑓𝑤) = (𝑃𝑤))
21eqeq1d 2198 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = ∅ ↔ (𝑃𝑤) = ∅))
32rexbidv 2491 . . . 4 (𝑓 = 𝑃 → (∃𝑤𝑂 (𝑓𝑤) = ∅ ↔ ∃𝑤𝑂 (𝑃𝑤) = ∅))
41eqeq1d 2198 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = 1o ↔ (𝑃𝑤) = 1o))
54ralbidv 2490 . . . 4 (𝑓 = 𝑃 → (∀𝑤𝑂 (𝑓𝑤) = 1o ↔ ∀𝑤𝑂 (𝑃𝑤) = 1o))
63, 5orbi12d 794 . . 3 (𝑓 = 𝑃 → ((∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o) ↔ (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o)))
7 fodjuomni.o . . . 4 (𝜑𝑂 ∈ Omni)
8 isomnimap 7170 . . . . 5 (𝑂 ∈ Omni → (𝑂 ∈ Omni ↔ ∀𝑓 ∈ (2o𝑚 𝑂)(∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o)))
97, 8syl 14 . . . 4 (𝜑 → (𝑂 ∈ Omni ↔ ∀𝑓 ∈ (2o𝑚 𝑂)(∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o)))
107, 9mpbid 147 . . 3 (𝜑 → ∀𝑓 ∈ (2o𝑚 𝑂)(∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o))
11 fodjuomni.fo . . . 4 (𝜑𝐹:𝑂onto→(𝐴𝐵))
12 fodjuomni.p . . . 4 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
1311, 12, 7fodjuf 7178 . . 3 (𝜑𝑃 ∈ (2o𝑚 𝑂))
146, 10, 13rspcdva 2861 . 2 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o))
1511adantr 276 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → 𝐹:𝑂onto→(𝐴𝐵))
16 simpr 110 . . . . . 6 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑤𝑂 (𝑃𝑤) = ∅)
17 fveqeq2 5546 . . . . . . 7 (𝑤 = 𝑣 → ((𝑃𝑤) = ∅ ↔ (𝑃𝑣) = ∅))
1817cbvrexv 2719 . . . . . 6 (∃𝑤𝑂 (𝑃𝑤) = ∅ ↔ ∃𝑣𝑂 (𝑃𝑣) = ∅)
1916, 18sylib 122 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑣𝑂 (𝑃𝑣) = ∅)
2015, 12, 19fodjum 7179 . . . 4 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑥 𝑥𝐴)
2120ex 115 . . 3 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ → ∃𝑥 𝑥𝐴))
2211adantr 276 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐹:𝑂onto→(𝐴𝐵))
23 simpr 110 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → ∀𝑤𝑂 (𝑃𝑤) = 1o)
2422, 12, 23fodju0 7180 . . . 4 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐴 = ∅)
2524ex 115 . . 3 (𝜑 → (∀𝑤𝑂 (𝑃𝑤) = 1o𝐴 = ∅))
2621, 25orim12d 787 . 2 (𝜑 → ((∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o) → (∃𝑥 𝑥𝐴𝐴 = ∅)))
2714, 26mpd 13 1 (𝜑 → (∃𝑥 𝑥𝐴𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  c0 3437  ifcif 3549  cmpt 4082  ontowfo 5236  cfv 5238  (class class class)co 5900  1oc1o 6438  2oc2o 6439  𝑚 cmap 6678  cdju 7070  inlcinl 7078  Omnicomni 7167
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-1o 6445  df-2o 6446  df-map 6680  df-dju 7071  df-inl 7080  df-inr 7081  df-omni 7168
This theorem is referenced by:  fodjuomni  7182
  Copyright terms: Public domain W3C validator