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Theorem fodjuomnilemres 6891
Description: Lemma for fodjuomni 6892. 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 5339 . . . . . 6 (𝑓 = 𝑃 → (𝑓𝑤) = (𝑃𝑤))
21eqeq1d 2103 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = ∅ ↔ (𝑃𝑤) = ∅))
32rexbidv 2392 . . . 4 (𝑓 = 𝑃 → (∃𝑤𝑂 (𝑓𝑤) = ∅ ↔ ∃𝑤𝑂 (𝑃𝑤) = ∅))
41eqeq1d 2103 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = 1o ↔ (𝑃𝑤) = 1o))
54ralbidv 2391 . . . 4 (𝑓 = 𝑃 → (∀𝑤𝑂 (𝑓𝑤) = 1o ↔ ∀𝑤𝑂 (𝑃𝑤) = 1o))
63, 5orbi12d 745 . . 3 (𝑓 = 𝑃 → ((∃𝑤𝑂 (𝑓𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑓𝑤) = 1o) ↔ (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o)))
7 fodjuomni.o . . . 4 (𝜑𝑂 ∈ Omni)
8 isomnimap 6880 . . . . 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 6888 . . 3 (𝜑𝑃 ∈ (2o𝑚 𝑂))
146, 10, 13rspcdva 2741 . 2 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ ∨ ∀𝑤𝑂 (𝑃𝑤) = 1o))
1511adantr 271 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → 𝐹:𝑂onto→(𝐴𝐵))
16 simpr 109 . . . . . 6 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑤𝑂 (𝑃𝑤) = ∅)
17 fveqeq2 5349 . . . . . . 7 (𝑤 = 𝑣 → ((𝑃𝑤) = ∅ ↔ (𝑃𝑣) = ∅))
1817cbvrexv 2605 . . . . . 6 (∃𝑤𝑂 (𝑃𝑤) = ∅ ↔ ∃𝑣𝑂 (𝑃𝑣) = ∅)
1916, 18sylib 121 . . . . 5 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑣𝑂 (𝑃𝑣) = ∅)
2015, 12, 19fodjum 6889 . . . 4 ((𝜑 ∧ ∃𝑤𝑂 (𝑃𝑤) = ∅) → ∃𝑥 𝑥𝐴)
2120ex 114 . . 3 (𝜑 → (∃𝑤𝑂 (𝑃𝑤) = ∅ → ∃𝑥 𝑥𝐴))
2211adantr 271 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐹:𝑂onto→(𝐴𝐵))
23 simpr 109 . . . . 5 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → ∀𝑤𝑂 (𝑃𝑤) = 1o)
2422, 12, 23fodju0 6890 . . . 4 ((𝜑 ∧ ∀𝑤𝑂 (𝑃𝑤) = 1o) → 𝐴 = ∅)
2524ex 114 . . 3 (𝜑 → (∀𝑤𝑂 (𝑃𝑤) = 1o𝐴 = ∅))
2621, 25orim12d 738 . 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 667   = wceq 1296  wex 1433  wcel 1445  wral 2370  wrex 2371  c0 3302  ifcif 3413  cmpt 3921  ontowfo 5047  cfv 5049  (class class class)co 5690  1oc1o 6212  2oc2o 6213  𝑚 cmap 6445  cdju 6810  inlcinl 6817  Omnicomni 6875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-1o 6219  df-2o 6220  df-map 6447  df-dju 6811  df-inl 6819  df-inr 6820  df-omni 6877
This theorem is referenced by:  fodjuomni  6892
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