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Theorem fodjuomnilemdc 7087
Description: Lemma for fodjuomni 7092. 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 5392 . . . . . 6 (𝐹:𝑂onto→(𝐴𝐵) → 𝐹:𝑂⟶(𝐴𝐵))
31, 2syl 14 . . . . 5 (𝜑𝐹:𝑂⟶(𝐴𝐵))
43ffvelrnda 5602 . . . 4 ((𝜑𝑋𝑂) → (𝐹𝑋) ∈ (𝐴𝐵))
5 djur 7013 . . . 4 ((𝐹𝑋) ∈ (𝐴𝐵) ↔ (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
64, 5sylib 121 . . 3 ((𝜑𝑋𝑂) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
7 nfv 1508 . . . . . . . 8 𝑧(𝜑𝑋𝑂)
8 nfre1 2500 . . . . . . . 8 𝑧𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)
97, 8nfan 1545 . . . . . . 7 𝑧((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
10 simpr 109 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
11 fveq2 5468 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
1211eqeq2d 2169 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑋) = (inr‘𝑧) ↔ (𝐹𝑋) = (inr‘𝑤)))
1312cbvrexv 2681 . . . . . . . . . 10 (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) ↔ ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
1410, 13sylib 121 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
15 vex 2715 . . . . . . . . . . . . . . 15 𝑧 ∈ V
16 vex 2715 . . . . . . . . . . . . . . 15 𝑤 ∈ V
17 djune 7022 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
1815, 16, 17mp2an 423 . . . . . . . . . . . . . 14 (inl‘𝑧) ≠ (inr‘𝑤)
19 neeq2 2341 . . . . . . . . . . . . . 14 ((𝐹𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
2018, 19mpbiri 167 . . . . . . . . . . . . 13 ((𝐹𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹𝑋))
2120necomd 2413 . . . . . . . . . . . 12 ((𝐹𝑋) = (inr‘𝑤) → (𝐹𝑋) ≠ (inl‘𝑧))
2221neneqd 2348 . . . . . . . . . . 11 ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧))
2322a1i 9 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2423rexlimdvw 2578 . . . . . . . . 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 2528 . . . . . 6 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧))
28 ralnex 2445 . . . . . 6 (∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧) ↔ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
2927, 28sylib 121 . . . . 5 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
3029ex 114 . . . 4 ((𝜑𝑋𝑂) → (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3130orim2d 778 . . 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 821 . 2 (DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ↔ (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3432, 33sylibr 133 1 ((𝜑𝑋𝑂) → DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  DECID wdc 820   = wceq 1335  wcel 2128  wne 2327  wral 2435  wrex 2436  Vcvv 2712  wf 5166  ontowfo 5168  cfv 5170  cdju 6981  inlcinl 6989  inrcinr 6990
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-1o 6363  df-dju 6982  df-inl 6991  df-inr 6992
This theorem is referenced by:  fodjuf  7088  fodjum  7089  fodju0  7090
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