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Theorem fodjuomnilemres 7020
 Description: Lemma for fodjuomni 7021. 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 5420 . . . . . 6 (𝑓 = 𝑃 → (𝑓𝑤) = (𝑃𝑤))
21eqeq1d 2148 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = ∅ ↔ (𝑃𝑤) = ∅))
32rexbidv 2438 . . . 4 (𝑓 = 𝑃 → (∃𝑤𝑂 (𝑓𝑤) = ∅ ↔ ∃𝑤𝑂 (𝑃𝑤) = ∅))
41eqeq1d 2148 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = 1o ↔ (𝑃𝑤) = 1o))
54ralbidv 2437 . . . 4 (𝑓 = 𝑃 → (∀𝑤𝑂 (𝑓𝑤) = 1o ↔ ∀𝑤𝑂 (𝑃𝑤) = 1o))
63, 5orbi12d 782 . . 3 (𝑓 = 𝑃 → ((∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o) ↔ (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o)))
7 fodjuomni.o . . . 4 (𝜑𝑂 ∈ Omni)
8 isomnimap 7009 . . . . 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 146 . . 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 7017 . . 3 (𝜑𝑃 ∈ (2o𝑚 𝑂))
146, 10, 13rspcdva 2794 . 2 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o))
1511adantr 274 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → 𝐹:𝑂onto→(𝐴𝐵))
16 simpr 109 . . . . . 6 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑤𝑂 (𝑃𝑤) = ∅)
17 fveqeq2 5430 . . . . . . 7 (𝑤 = 𝑣 → ((𝑃𝑤) = ∅ ↔ (𝑃𝑣) = ∅))
1817cbvrexv 2655 . . . . . 6 (∃𝑤𝑂 (𝑃𝑤) = ∅ ↔ ∃𝑣𝑂 (𝑃𝑣) = ∅)
1916, 18sylib 121 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑣𝑂 (𝑃𝑣) = ∅)
2015, 12, 19fodjum 7018 . . . 4 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑥 𝑥𝐴)
2120ex 114 . . 3 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ → ∃𝑥 𝑥𝐴))
2211adantr 274 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐹:𝑂onto→(𝐴𝐵))
23 simpr 109 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → ∀𝑤𝑂 (𝑃𝑤) = 1o)
2422, 12, 23fodju0 7019 . . . 4 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐴 = ∅)
2524ex 114 . . 3 (𝜑 → (∀𝑤𝑂 (𝑃𝑤) = 1o𝐴 = ∅))
2621, 25orim12d 775 . 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 103   ↔ wb 104   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ∅c0 3363  ifcif 3474   ↦ cmpt 3989  –onto→wfo 5121  ‘cfv 5123  (class class class)co 5774  1oc1o 6306  2oc2o 6307   ↑𝑚 cmap 6542   ⊔ cdju 6922  inlcinl 6930  Omnicomni 7004 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-1o 6313  df-2o 6314  df-map 6544  df-dju 6923  df-inl 6932  df-inr 6933  df-omni 7006 This theorem is referenced by:  fodjuomni  7021
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