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Theorem fodjum 7175
Description: Lemma for fodjuomni 7178 and fodjumkv 7189. 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 6458 . . . . . . . . 9 1o ≠ ∅
32nesymi 2406 . . . . . . . 8 ¬ ∅ = 1o
43intnan 930 . . . . . . 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 5543 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝐹𝑦) = (inl‘𝑧) ↔ (𝐹𝑤) = (inl‘𝑧)))
98rexbidv 2491 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑧𝐴 (𝐹𝑦) = (inl‘𝑧) ↔ ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧)))
109ifbid 3570 . . . . . . . . 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 4611 . . . . . . . . . . 11 ∅ ∈ ω
1312a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ ∈ ω)
14 1onn 6546 . . . . . . . . . . 11 1o ∈ ω
1514a1i 9 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → 1o ∈ ω)
16 fodjuf.fo . . . . . . . . . . . 12 (𝜑𝐹:𝑂onto→(𝐴𝐵))
1716fodjuomnilemdc 7173 . . . . . . . . . . 11 ((𝜑𝑤𝑂) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1817adantrr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → DECID𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
1913, 15, 18ifcldcd 3585 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o) ∈ ω)
207, 10, 11, 19fvmptd3 5630 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (𝑃𝑤) = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
216, 20eqtr3d 2224 . . . . . . 7 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∅ = if(∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧), ∅, 1o))
22 eqifdc 3584 . . . . . . . 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 1360 . . . . 5 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) ∧ ∅ = ∅))
2625simpld 112 . . . 4 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧))
27 rexm 3537 . . . 4 (∃𝑧𝐴 (𝐹𝑤) = (inl‘𝑧) → ∃𝑧 𝑧𝐴)
2826, 27syl 14 . . 3 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑧 𝑧𝐴)
29 eleq1w 2250 . . . 4 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
3029cbvexv 1930 . . 3 (∃𝑧 𝑧𝐴 ↔ ∃𝑥 𝑥𝐴)
3128, 30sylib 122 . 2 ((𝜑 ∧ (𝑤𝑂 ∧ (𝑃𝑤) = ∅)) → ∃𝑥 𝑥𝐴)
321, 31rexlimddv 2612 1 (𝜑 → ∃𝑥 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  wex 1503  wcel 2160  wrex 2469  c0 3437  ifcif 3549  cmpt 4079  ωcom 4607  ontowfo 5233  cfv 5235  1oc1o 6435  cdju 7067  inlcinl 7075
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-dju 7068  df-inl 7077  df-inr 7078
This theorem is referenced by:  fodjuomnilemres  7177  fodjumkvlemres  7188
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