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

Theorem fodjum 7134
Description: Lemma for fodjuomni 7137 and fodjumkv 7148. 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 6423 . . . . . . . . 9 1o ≠ ∅
32nesymi 2391 . . . . . . . 8 ¬ ∅ = 1o
43intnan 929 . . . . . . 7 ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o)
54a1i 9 . . . . . 6 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o))
6 simprr 531 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = ∅)
7 fodjuf.p . . . . . . . . 9 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
8 fveqeq2 5516 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑤) = (inl‘𝑧)))
98rexbidv 2476 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧)))
109ifbid 3553 . . . . . . . . 9 (𝑦 = 𝑤 → if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
11 simprl 529 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 𝑤𝑂)
12 peano1 4587 . . . . . . . . . . 11 ∅ ∈ ω
1312a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ ∈ ω)
14 1onn 6511 . . . . . . . . . . 11 1o ∈ ω
1514a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 1o ∈ ω)
16 fodjuf.fo . . . . . . . . . . . 12 (𝜑𝐹:𝑂onto→(𝐴𝐵))
1716fodjuomnilemdc 7132 . . . . . . . . . . 11 ((𝜑𝑤𝑂) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1817adantrr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1913, 15, 18ifcldcd 3567 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
207, 10, 11, 19fvmptd3 5601 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
216, 20eqtr3d 2210 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
22 eqifdc 3566 . . . . . . . 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 147 . . . . . 6 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ((∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅) ∨ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o)))
255, 24ecased 1349 . . . . 5 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
2625simpld 112 . . . 4 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
27 rexm 3520 . . . 4 (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → ∃𝑧 𝑧𝐴)
2826, 27syl 14 . . 3 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧 𝑧𝐴)
29 eleq1w 2236 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
3029cbvexv 1916 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑥 𝑥𝐴)
3128, 30sylib 122 . 2 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑥 𝑥𝐴)
321, 31rexlimddv 2597 1 (𝜑 → ∃𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wex 1490  wcel 2146  wrex 2454  c0 3420  ifcif 3532  cmpt 4059  ωcom 4583  ontowfo 5206  cfv 5208  1oc1o 6400  cdju 7026  inlcinl 7034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037
This theorem is referenced by:  fodjuomnilemres  7136  fodjumkvlemres  7147
  Copyright terms: Public domain W3C validator