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Theorem fodjumkvlemres 7115
Description: Lemma for fodjumkv 7116. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjumkv.o (𝜑𝑀 ∈ Markov)
fodjumkv.fo (𝜑𝐹:𝑀onto→(𝐴𝐵))
fodjumkv.p 𝑃 = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
Assertion
Ref Expression
fodjumkvlemres (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))
Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑀,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)   𝑀(𝑥)

Proof of Theorem fodjumkvlemres
Dummy variables 𝑣 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjumkv.fo . . . . . 6 (𝜑𝐹:𝑀onto→(𝐴𝐵))
21adantr 274 . . . . 5 ((𝜑 ∧ ∀𝑤𝑀 (𝑃𝑤) = 1o) → 𝐹:𝑀onto→(𝐴𝐵))
3 fodjumkv.p . . . . 5 𝑃 = (𝑦𝑀 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
4 simpr 109 . . . . 5 ((𝜑 ∧ ∀𝑤𝑀 (𝑃𝑤) = 1o) → ∀𝑤𝑀 (𝑃𝑤) = 1o)
52, 3, 4fodju0 7103 . . . 4 ((𝜑 ∧ ∀𝑤𝑀 (𝑃𝑤) = 1o) → 𝐴 = ∅)
65ex 114 . . 3 (𝜑 → (∀𝑤𝑀 (𝑃𝑤) = 1o𝐴 = ∅))
76necon3ad 2376 . 2 (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤𝑀 (𝑃𝑤) = 1o))
8 fveq1 5480 . . . . . . 7 (𝑓 = 𝑃 → (𝑓𝑤) = (𝑃𝑤))
98eqeq1d 2173 . . . . . 6 (𝑓 = 𝑃 → ((𝑓𝑤) = 1o ↔ (𝑃𝑤) = 1o))
109ralbidv 2464 . . . . 5 (𝑓 = 𝑃 → (∀𝑤𝑀 (𝑓𝑤) = 1o ↔ ∀𝑤𝑀 (𝑃𝑤) = 1o))
1110notbid 657 . . . 4 (𝑓 = 𝑃 → (¬ ∀𝑤𝑀 (𝑓𝑤) = 1o ↔ ¬ ∀𝑤𝑀 (𝑃𝑤) = 1o))
128eqeq1d 2173 . . . . 5 (𝑓 = 𝑃 → ((𝑓𝑤) = ∅ ↔ (𝑃𝑤) = ∅))
1312rexbidv 2465 . . . 4 (𝑓 = 𝑃 → (∃𝑤𝑀 (𝑓𝑤) = ∅ ↔ ∃𝑤𝑀 (𝑃𝑤) = ∅))
1411, 13imbi12d 233 . . 3 (𝑓 = 𝑃 → ((¬ ∀𝑤𝑀 (𝑓𝑤) = 1o → ∃𝑤𝑀 (𝑓𝑤) = ∅) ↔ (¬ ∀𝑤𝑀 (𝑃𝑤) = 1o → ∃𝑤𝑀 (𝑃𝑤) = ∅)))
15 fodjumkv.o . . . 4 (𝜑𝑀 ∈ Markov)
16 ismkvmap 7110 . . . . 5 (𝑀 ∈ Markov → (𝑀 ∈ Markov ↔ ∀𝑓 ∈ (2o𝑚 𝑀)(¬ ∀𝑤𝑀 (𝑓𝑤) = 1o → ∃𝑤𝑀 (𝑓𝑤) = ∅)))
1716ibi 175 . . . 4 (𝑀 ∈ Markov → ∀𝑓 ∈ (2o𝑚 𝑀)(¬ ∀𝑤𝑀 (𝑓𝑤) = 1o → ∃𝑤𝑀 (𝑓𝑤) = ∅))
1815, 17syl 14 . . 3 (𝜑 → ∀𝑓 ∈ (2o𝑚 𝑀)(¬ ∀𝑤𝑀 (𝑓𝑤) = 1o → ∃𝑤𝑀 (𝑓𝑤) = ∅))
191, 3, 15fodjuf 7101 . . 3 (𝜑𝑃 ∈ (2o𝑚 𝑀))
2014, 18, 19rspcdva 2831 . 2 (𝜑 → (¬ ∀𝑤𝑀 (𝑃𝑤) = 1o → ∃𝑤𝑀 (𝑃𝑤) = ∅))
211adantr 274 . . . 4 ((𝜑 ∧ ∃𝑤𝑀 (𝑃𝑤) = ∅) → 𝐹:𝑀onto→(𝐴𝐵))
22 simpr 109 . . . . 5 ((𝜑 ∧ ∃𝑤𝑀 (𝑃𝑤) = ∅) → ∃𝑤𝑀 (𝑃𝑤) = ∅)
23 fveqeq2 5490 . . . . . 6 (𝑤 = 𝑣 → ((𝑃𝑤) = ∅ ↔ (𝑃𝑣) = ∅))
2423cbvrexv 2691 . . . . 5 (∃𝑤𝑀 (𝑃𝑤) = ∅ ↔ ∃𝑣𝑀 (𝑃𝑣) = ∅)
2522, 24sylib 121 . . . 4 ((𝜑 ∧ ∃𝑤𝑀 (𝑃𝑤) = ∅) → ∃𝑣𝑀 (𝑃𝑣) = ∅)
2621, 3, 25fodjum 7102 . . 3 ((𝜑 ∧ ∃𝑤𝑀 (𝑃𝑤) = ∅) → ∃𝑥 𝑥𝐴)
2726ex 114 . 2 (𝜑 → (∃𝑤𝑀 (𝑃𝑤) = ∅ → ∃𝑥 𝑥𝐴))
287, 20, 273syld 57 1 (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1342  wex 1479  wcel 2135  wne 2334  wral 2442  wrex 2443  c0 3405  ifcif 3516  cmpt 4038  ontowfo 5181  cfv 5183  (class class class)co 5837  1oc1o 6369  2oc2o 6370  𝑚 cmap 6606  cdju 6994  inlcinl 7002  Markovcmarkov 7107
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-if 3517  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-1o 6376  df-2o 6377  df-map 6608  df-dju 6995  df-inl 7004  df-inr 7005  df-markov 7108
This theorem is referenced by:  fodjumkv  7116
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