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Theorem fodjum 7138
Description: Lemma for fodjuomni 7141 and fodjumkv 7152. 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 6427 . . . . . . . . 9 1o ≠ ∅
32nesymi 2393 . . . . . . . 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 5520 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑤) = (inl‘𝑧)))
98rexbidv 2478 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧)))
109ifbid 3555 . . . . . . . . 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 4590 . . . . . . . . . . 11 ∅ ∈ ω
1312a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ ∈ ω)
14 1onn 6515 . . . . . . . . . . 11 1o ∈ ω
1514a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 1o ∈ ω)
16 fodjuf.fo . . . . . . . . . . . 12 (𝜑𝐹:𝑂onto→(𝐴𝐵))
1716fodjuomnilemdc 7136 . . . . . . . . . . 11 ((𝜑𝑤𝑂) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1817adantrr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1913, 15, 18ifcldcd 3569 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
207, 10, 11, 19fvmptd3 5605 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
216, 20eqtr3d 2212 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
22 eqifdc 3568 . . . . . . . 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 3522 . . . 4 (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → ∃𝑧 𝑧𝐴)
2826, 27syl 14 . . 3 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧 𝑧𝐴)
29 eleq1w 2238 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
3029cbvexv 1918 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑥 𝑥𝐴)
3128, 30sylib 122 . 2 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑥 𝑥𝐴)
321, 31rexlimddv 2599 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 1492  wcel 2148  wrex 2456  c0 3422  ifcif 3534  cmpt 4061  ωcom 4586  ontowfo 5210  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-dju 7031  df-inl 7040  df-inr 7041
This theorem is referenced by:  fodjuomnilemres  7140  fodjumkvlemres  7151
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