ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fodjum GIF version

Theorem fodjum 6930
Description: Lemma for fodjuomni 6933 and fodjumkv 6945. A condition which shows that 𝐴 is inhabited. (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))
fodjum.z (𝜑 → ∃𝑤𝑂 (𝑃𝑤) = ∅)
Assertion
Ref Expression
fodjum (𝜑 → ∃𝑥 𝑥𝐴)
Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑤,𝐴,𝑥,𝑧   𝑦,𝐴,𝑤   𝑦,𝐹   𝜑,𝑤
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑦,𝑤)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑥,𝑤)   𝑂(𝑥,𝑤)

Proof of Theorem fodjum
StepHypRef Expression
1 fodjum.z . 2 (𝜑 → ∃𝑤𝑂 (𝑃𝑤) = ∅)
2 1n0 6259 . . . . . . . . 9 1o ≠ ∅
32nesymi 2313 . . . . . . . 8 ¬ ∅ = 1o
43intnan 882 . . . . . . 7 ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o)
54a1i 9 . . . . . 6 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o))
6 simprr 502 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = ∅)
7 fodjuf.p . . . . . . . . 9 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
8 fveqeq2 5362 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑤) = (inl‘𝑧)))
98rexbidv 2397 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧)))
109ifbid 3440 . . . . . . . . 9 (𝑦 = 𝑤 → if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
11 simprl 501 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 𝑤𝑂)
12 peano1 4446 . . . . . . . . . . 11 ∅ ∈ ω
1312a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ ∈ ω)
14 1onn 6346 . . . . . . . . . . 11 1o ∈ ω
1514a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 1o ∈ ω)
16 fodjuf.fo . . . . . . . . . . . 12 (𝜑𝐹:𝑂onto→(𝐴𝐵))
1716fodjuomnilemdc 6928 . . . . . . . . . . 11 ((𝜑𝑤𝑂) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1817adantrr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1913, 15, 18ifcldcd 3454 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
207, 10, 11, 19fvmptd3 5446 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
216, 20eqtr3d 2134 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
22 eqifdc 3453 . . . . . . . 8 (DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → (∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ↔ ((∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o))))
2318, 22syl 14 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ↔ ((∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o))))
2421, 23mpbid 146 . . . . . 6 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ((∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o)))
255, 24ecased 1295 . . . . 5 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
2625simpld 111 . . . 4 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
27 rexm 3409 . . . 4 (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → ∃𝑧 𝑧𝐴)
2826, 27syl 14 . . 3 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧 𝑧𝐴)
29 eleq1w 2160 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
3029cbvexv 1855 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑥 𝑥𝐴)
3128, 30sylib 121 . 2 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑥 𝑥𝐴)
321, 31rexlimddv 2513 1 (𝜑 → ∃𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 670  DECID wdc 786   = wceq 1299  wex 1436  wcel 1448  wrex 2376  c0 3310  ifcif 3421  cmpt 3929  ωcom 4442  ontowfo 5057  cfv 5059  1oc1o 6236  cdju 6837  inlcinl 6845
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1st 5969  df-2nd 5970  df-1o 6243  df-dju 6838  df-inl 6847  df-inr 6848
This theorem is referenced by:  fodjuomnilemres  6932  fodjumkvlemres  6944
  Copyright terms: Public domain W3C validator