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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  ontowfo 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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