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Theorem fodjuomnilemdc 7009
Description: Lemma for fodjuomni 7014. 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 5340 . . . . . 6 (𝐹:𝑂onto→(𝐴𝐵) → 𝐹:𝑂⟶(𝐴𝐵))
31, 2syl 14 . . . . 5 (𝜑𝐹:𝑂⟶(𝐴𝐵))
43ffvelrnda 5548 . . . 4 ((𝜑𝑋𝑂) → (𝐹𝑋) ∈ (𝐴𝐵))
5 djur 6947 . . . 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 2474 . . . . . . . 8 𝑧𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)
97, 8nfan 1544 . . . . . . 7 𝑧((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
10 simpr 109 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
11 fveq2 5414 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
1211eqeq2d 2149 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑋) = (inr‘𝑧) ↔ (𝐹𝑋) = (inr‘𝑤)))
1312cbvrexv 2653 . . . . . . . . . 10 (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) ↔ ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
1410, 13sylib 121 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
15 vex 2684 . . . . . . . . . . . . . . 15 𝑧 ∈ V
16 vex 2684 . . . . . . . . . . . . . . 15 𝑤 ∈ V
17 djune 6956 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
1815, 16, 17mp2an 422 . . . . . . . . . . . . . 14 (inl‘𝑧) ≠ (inr‘𝑤)
19 neeq2 2320 . . . . . . . . . . . . . 14 ((𝐹𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
2018, 19mpbiri 167 . . . . . . . . . . . . 13 ((𝐹𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹𝑋))
2120necomd 2392 . . . . . . . . . . . 12 ((𝐹𝑋) = (inr‘𝑤) → (𝐹𝑋) ≠ (inl‘𝑧))
2221neneqd 2327 . . . . . . . . . . 11 ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧))
2322a1i 9 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2423rexlimdvw 2551 . . . . . . . . 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 2501 . . . . . 6 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧))
28 ralnex 2424 . . . . . 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 777 . . 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 820 . 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 697  DECID wdc 819   = wceq 1331  wcel 1480  wne 2306  wral 2414  wrex 2415  Vcvv 2681  wf 5114  ontowfo 5116  cfv 5118  cdju 6915  inlcinl 6923  inrcinr 6924
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926
This theorem is referenced by:  fodjuf  7010  fodjum  7011  fodju0  7012
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