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Theorem fodjum 7026
 Description: Lemma for fodjuomni 7029 and fodjumkv 7042. 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 6337 . . . . . . . . 9 1o ≠ ∅
32nesymi 2355 . . . . . . . 8 ¬ ∅ = 1o
43intnan 915 . . . . . . 7 ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o)
54a1i 9 . . . . . 6 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ¬ (¬ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = 1o))
6 simprr 522 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = ∅)
7 fodjuf.p . . . . . . . . 9 𝑃 = (𝑦𝑂 ↦ if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o))
8 fveqeq2 5438 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑤) = (inl‘𝑧)))
98rexbidv 2439 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧)))
109ifbid 3498 . . . . . . . . 9 (𝑦 = 𝑤 → if(∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧), ∅, 1o) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
11 simprl 521 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 𝑤𝑂)
12 peano1 4516 . . . . . . . . . . 11 ∅ ∈ ω
1312a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ ∈ ω)
14 1onn 6424 . . . . . . . . . . 11 1o ∈ ω
1514a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 1o ∈ ω)
16 fodjuf.fo . . . . . . . . . . . 12 (𝜑𝐹:𝑂onto→(𝐴𝐵))
1716fodjuomnilemdc 7024 . . . . . . . . . . 11 ((𝜑𝑤𝑂) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1817adantrr 471 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1913, 15, 18ifcldcd 3512 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
207, 10, 11, 19fvmptd3 5522 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
216, 20eqtr3d 2175 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
22 eqifdc 3511 . . . . . . . 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 1328 . . . . 5 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
2625simpld 111 . . . 4 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
27 rexm 3467 . . . 4 (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → ∃𝑧 𝑧𝐴)
2826, 27syl 14 . . 3 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧 𝑧𝐴)
29 eleq1w 2201 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
3029cbvexv 1891 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑥 𝑥𝐴)
3128, 30sylib 121 . 2 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑥 𝑥𝐴)
321, 31rexlimddv 2557 1 (𝜑 → ∃𝑥 𝑥𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418  ∅c0 3368  ifcif 3479   ↦ cmpt 3997  ωcom 4512  –onto→wfo 5129  ‘cfv 5131  1oc1o 6314   ⊔ cdju 6930  inlcinl 6938 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 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363 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 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-dju 6931  df-inl 6940  df-inr 6941 This theorem is referenced by:  fodjuomnilemres  7028  fodjumkvlemres  7041
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