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Theorem fodjuomnilemdc 7160
Description: Lemma for fodjuomni 7165. 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 5453 . . . . . 6 (𝐹:𝑂onto→(𝐴𝐵) → 𝐹:𝑂⟶(𝐴𝐵))
31, 2syl 14 . . . . 5 (𝜑𝐹:𝑂⟶(𝐴𝐵))
43ffvelcdmda 5667 . . . 4 ((𝜑𝑋𝑂) → (𝐹𝑋) ∈ (𝐴𝐵))
5 djur 7086 . . . 4 ((𝐹𝑋) ∈ (𝐴𝐵) ↔ (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
64, 5sylib 122 . . 3 ((𝜑𝑋𝑂) → (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)))
7 nfv 1539 . . . . . . . 8 𝑧(𝜑𝑋𝑂)
8 nfre1 2533 . . . . . . . 8 𝑧𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)
97, 8nfan 1576 . . . . . . 7 𝑧((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
10 simpr 110 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
11 fveq2 5530 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
1211eqeq2d 2201 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑋) = (inr‘𝑧) ↔ (𝐹𝑋) = (inr‘𝑤)))
1312cbvrexv 2719 . . . . . . . . . 10 (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) ↔ ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
1410, 13sylib 122 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
15 vex 2755 . . . . . . . . . . . . . . 15 𝑧 ∈ V
16 vex 2755 . . . . . . . . . . . . . . 15 𝑤 ∈ V
17 djune 7095 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
1815, 16, 17mp2an 426 . . . . . . . . . . . . . 14 (inl‘𝑧) ≠ (inr‘𝑤)
19 neeq2 2374 . . . . . . . . . . . . . 14 ((𝐹𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
2018, 19mpbiri 168 . . . . . . . . . . . . 13 ((𝐹𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹𝑋))
2120necomd 2446 . . . . . . . . . . . 12 ((𝐹𝑋) = (inr‘𝑤) → (𝐹𝑋) ≠ (inl‘𝑧))
2221neneqd 2381 . . . . . . . . . . 11 ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧))
2322a1i 9 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2423rexlimdvw 2611 . . . . . . . . 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 2561 . . . . . 6 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧))
28 ralnex 2478 . . . . . 6 (∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧) ↔ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
2927, 28sylib 122 . . . . 5 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
3029ex 115 . . . 4 ((𝜑𝑋𝑂) → (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) → ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3130orim2d 789 . . 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 836 . 2 (DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ↔ (∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧𝐴 (𝐹𝑋) = (inl‘𝑧)))
3432, 33sylibr 134 1 ((𝜑𝑋𝑂) → DECID𝑧𝐴 (𝐹𝑋) = (inl‘𝑧))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  DECID wdc 835   = wceq 1364  wcel 2160  wne 2360  wral 2468  wrex 2469  Vcvv 2752  wf 5227  ontowfo 5229  cfv 5231  cdju 7054  inlcinl 7062  inrcinr 7063
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 4189  ax-pr 4224  ax-un 4448
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-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-1st 6159  df-2nd 6160  df-1o 6435  df-dju 7055  df-inl 7064  df-inr 7065
This theorem is referenced by:  fodjuf  7161  fodjum  7162  fodju0  7163
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