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Theorem fodjuomnilemdc 6743
Description: Lemma for fodjuomni 6748. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
Hypothesis
Ref Expression
fodjuomnilemdc.fo (𝜑𝐹:𝑂onto→(𝐴𝐵))
Assertion
Ref Expression
fodjuomnilemdc ((𝜑𝑋𝑂) → DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑧,𝑂   𝑧,𝑋   𝜑,𝑧

Proof of Theorem fodjuomnilemdc
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fodjuomnilemdc.fo . . . . . 6 (𝜑𝐹:𝑂onto→(𝐴𝐵))
2 fof 5196 . . . . . 6 (𝐹:𝑂onto→(𝐴𝐵) → 𝐹:𝑂⟶(𝐴𝐵))
31, 2syl 14 . . . . 5 (𝜑𝐹:𝑂⟶(𝐴𝐵))
43ffvelrnda 5397 . . . 4 ((𝜑𝑋𝑂) → (𝐹𝑋) ∈ (𝐴𝐵))
5 djur 6701 . . . 4 ((𝐹𝑋) ∈ (𝐴𝐵) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
64, 5syl 14 . . 3 ((𝜑𝑋𝑂) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
7 nfv 1464 . . . . . . . 8 𝑧(𝜑𝑋𝑂)
8 nfre1 2415 . . . . . . . 8 𝑧𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)
97, 8nfan 1500 . . . . . . 7 𝑧((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
10 simpr 108 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
11 fveq2 5268 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
1211eqeq2d 2096 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑋) = (inr‘𝑧) ↔ (𝐹𝑋) = (inr‘𝑤)))
1312cbvrexv 2587 . . . . . . . . . 10 (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) ↔ ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
1410, 13sylib 120 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
15 vex 2618 . . . . . . . . . . . . . . 15 𝑧 ∈ V
16 vex 2618 . . . . . . . . . . . . . . 15 𝑤 ∈ V
17 djune 6713 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
1815, 16, 17mp2an 417 . . . . . . . . . . . . . 14 (inl‘𝑧) ≠ (inr‘𝑤)
19 neeq2 2265 . . . . . . . . . . . . . 14 ((𝐹𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
2018, 19mpbiri 166 . . . . . . . . . . . . 13 ((𝐹𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹𝑋))
2120necomd 2337 . . . . . . . . . . . 12 ((𝐹𝑋) = (inr‘𝑤) → (𝐹𝑋) ≠ (inl‘𝑧))
2221neneqd 2272 . . . . . . . . . . 11 ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧))
2322a1i 9 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2423rexlimdvw 2488 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → (∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2514, 24mpd 13 . . . . . . . 8 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ¬ (𝐹𝑋) = (inl‘𝑧))
2625a1d 22 . . . . . . 7 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → (𝑧𝐴 → ¬ (𝐹𝑋) = (inl‘𝑧)))
279, 26ralrimi 2440 . . . . . 6 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧))
28 ralnex 2365 . . . . . 6 (∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧) ↔ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
2927, 28sylib 120 . . . . 5 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
3029ex 113 . . . 4 ((𝜑𝑋𝑂) → (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3130orim2d 735 . . 3 ((𝜑𝑋𝑂) → ((∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))))
326, 31mpd 13 . 2 ((𝜑𝑋𝑂) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
33 df-dc 779 . 2 (DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ↔ (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3432, 33sylibr 132 1 ((𝜑𝑋𝑂) → DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  DECID wdc 778   = wceq 1287  wcel 1436  wne 2251  wral 2355  wrex 2356  Vcvv 2615  wf 4977  ontowfo 4979  cfv 4981  cdju 6674  inlcinl 6681  inrcinr 6682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fo 4987  df-fv 4989  df-1st 5868  df-2nd 5869  df-1o 6135  df-dju 6675  df-inl 6683  df-inr 6684
This theorem is referenced by:  fodjuomnilemf  6744  fodjuomnilemm  6745  fodjuomnilem0  6746
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