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Theorem fodju0 7026
 Description: Lemma for fodjuomni 7028 and fodjumkv 7041. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
Hypotheses
Ref Expression
fodjuf.fo (𝜑𝐹:𝑂onto→(𝐴𝐵))
fodjuf.p 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
fodju0.1 (𝜑 → ∀𝑤𝑂 (𝑃𝑤) = 1o)
Assertion
Ref Expression
fodju0 (𝜑𝐴 = ∅)
Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑦,𝐴   𝑦,𝐹   𝑤,𝑂   𝑤,𝑃
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑦,𝑤)   𝑃(𝑦,𝑧)   𝐹(𝑤)

Proof of Theorem fodju0
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fodjuf.fo . . . . 5 (𝜑𝐹:𝑂onto→(𝐴𝐵))
2 djulcl 6943 . . . . 5 (𝑢𝐴 → (inl‘𝑢) ∈ (𝐴𝐵))
3 foelrn 5661 . . . . 5 ((𝐹:𝑂onto→(𝐴𝐵) ∧ (inl‘𝑢) ∈ (𝐴𝐵)) → ∃𝑣𝑂 (inl‘𝑢) = (𝐹𝑣))
41, 2, 3syl2an 287 . . . 4 ((𝜑𝑢𝐴) → ∃𝑣𝑂 (inl‘𝑢) = (𝐹𝑣))
5 fodjuf.p . . . . . 6 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
6 fveqeq2 5437 . . . . . . . 8 (𝑦 = 𝑣 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑣) = (inl‘𝑧)))
76rexbidv 2439 . . . . . . 7 (𝑦 = 𝑣 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧)))
87ifbid 3497 . . . . . 6 (𝑦 = 𝑣 → if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧), ∅, 1o))
9 simprl 521 . . . . . 6 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → 𝑣𝑂)
10 peano1 4515 . . . . . . . 8 ∅ ∈ ω
1110a1i 9 . . . . . . 7 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → ∅ ∈ ω)
12 1onn 6423 . . . . . . . 8 1o ∈ ω
1312a1i 9 . . . . . . 7 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → 1o ∈ ω)
141fodjuomnilemdc 7023 . . . . . . . 8 ((𝜑𝑣𝑂) → DECID𝑧𝐴 (𝐹𝑣) = (inl‘𝑧))
1514ad2ant2r 501 . . . . . . 7 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → DECID𝑧𝐴 (𝐹𝑣) = (inl‘𝑧))
1611, 13, 15ifcldcd 3511 . . . . . 6 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → if(∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧), ∅, 1o) ∈ ω)
175, 8, 9, 16fvmptd3 5521 . . . . 5 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → (𝑃𝑣) = if(∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧), ∅, 1o))
18 fveqeq2 5437 . . . . . 6 (𝑤 = 𝑣 → ((𝑃𝑤) = 1o ↔ (𝑃𝑣) = 1o))
19 fodju0.1 . . . . . . 7 (𝜑 → ∀𝑤𝑂 (𝑃𝑤) = 1o)
2019ad2antrr 480 . . . . . 6 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → ∀𝑤𝑂 (𝑃𝑤) = 1o)
2118, 20, 9rspcdva 2797 . . . . 5 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → (𝑃𝑣) = 1o)
22 simplr 520 . . . . . . 7 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → 𝑢𝐴)
23 simprr 522 . . . . . . . 8 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → (inl‘𝑢) = (𝐹𝑣))
2423eqcomd 2146 . . . . . . 7 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → (𝐹𝑣) = (inl‘𝑢))
25 fveq2 5428 . . . . . . . 8 (𝑧 = 𝑢 → (inl‘𝑧) = (inl‘𝑢))
2625rspceeqv 2810 . . . . . . 7 ((𝑢𝐴 ∧ (𝐹𝑣) = (inl‘𝑢)) → ∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧))
2722, 24, 26syl2anc 409 . . . . . 6 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → ∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧))
2827iftrued 3485 . . . . 5 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → if(∃𝑧𝐴 (𝐹𝑣) = (inl‘𝑧), ∅, 1o) = ∅)
2917, 21, 283eqtr3rd 2182 . . . 4 (((𝜑𝑢𝐴) ∧ (𝑣𝑂 ∧ (inl‘𝑢) = (𝐹𝑣))) → ∅ = 1o)
304, 29rexlimddv 2557 . . 3 ((𝜑𝑢𝐴) → ∅ = 1o)
31 1n0 6336 . . . . 5 1o ≠ ∅
3231nesymi 2355 . . . 4 ¬ ∅ = 1o
3332a1i 9 . . 3 ((𝜑𝑢𝐴) → ¬ ∅ = 1o)
3430, 33pm2.65da 651 . 2 (𝜑 → ¬ 𝑢𝐴)
3534eq0rdv 3411 1 (𝜑𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 820   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∅c0 3367  ifcif 3478   ↦ cmpt 3996  ωcom 4511  –onto→wfo 5128  ‘cfv 5130  1oc1o 6313   ⊔ cdju 6929  inlcinl 6937 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 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362 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-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-2nd 6046  df-1o 6320  df-dju 6930  df-inl 6939  df-inr 6940 This theorem is referenced by:  fodjuomnilemres  7027  fodjumkvlemres  7040
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