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Theorem fodjuomnilemdc 7120
Description: Lemma for fodjuomni 7125. 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 5420 . . . . . 6 (𝐹:𝑂onto→(𝐴𝐵) → 𝐹:𝑂⟶(𝐴𝐵))
31, 2syl 14 . . . . 5 (𝜑𝐹:𝑂⟶(𝐴𝐵))
43ffvelrnda 5631 . . . 4 ((𝜑𝑋𝑂) → (𝐹𝑋) ∈ (𝐴𝐵))
5 djur 7046 . . . 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 1521 . . . . . . . 8 𝑧(𝜑𝑋𝑂)
8 nfre1 2513 . . . . . . . 8 𝑧𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)
97, 8nfan 1558 . . . . . . 7 𝑧((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
10 simpr 109 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧))
11 fveq2 5496 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
1211eqeq2d 2182 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑋) = (inr‘𝑧) ↔ (𝐹𝑋) = (inr‘𝑤)))
1312cbvrexv 2697 . . . . . . . . . 10 (∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧) ↔ ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
1410, 13sylib 121 . . . . . . . . 9 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∃𝑤𝐵 (𝐹𝑋) = (inr‘𝑤))
15 vex 2733 . . . . . . . . . . . . . . 15 𝑧 ∈ V
16 vex 2733 . . . . . . . . . . . . . . 15 𝑤 ∈ V
17 djune 7055 . . . . . . . . . . . . . . 15 ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
1815, 16, 17mp2an 424 . . . . . . . . . . . . . 14 (inl‘𝑧) ≠ (inr‘𝑤)
19 neeq2 2354 . . . . . . . . . . . . . 14 ((𝐹𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
2018, 19mpbiri 167 . . . . . . . . . . . . 13 ((𝐹𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹𝑋))
2120necomd 2426 . . . . . . . . . . . 12 ((𝐹𝑋) = (inr‘𝑤) → (𝐹𝑋) ≠ (inl‘𝑧))
2221neneqd 2361 . . . . . . . . . . 11 ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧))
2322a1i 9 . . . . . . . . . 10 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ((𝐹𝑋) = (inr‘𝑤) → ¬ (𝐹𝑋) = (inl‘𝑧)))
2423rexlimdvw 2591 . . . . . . . . 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 2541 . . . . . 6 (((𝜑𝑋𝑂) ∧ ∃𝑧𝐵 (𝐹𝑋) = (inr‘𝑧)) → ∀𝑧𝐴 ¬ (𝐹𝑋) = (inl‘𝑧))
28 ralnex 2458 . . . . . 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 783 . . 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 830 . 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 703  DECID wdc 829   = wceq 1348  wcel 2141  wne 2340  wral 2448  wrex 2449  Vcvv 2730  wf 5194  ontowfo 5196  cfv 5198  cdju 7014  inlcinl 7022  inrcinr 7023
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025
This theorem is referenced by:  fodjuf  7121  fodjum  7122  fodju0  7123
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