Theorem fodjuomnilemdc 7155

Description: Lemma for fodjuomni 7160. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)

Hypothesis

Ref	Expression
fodjuomnilemdc.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))

Assertion

Ref	Expression
fodjuomnilemdc	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (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.

Step	Hyp	Ref	Expression
1		fodjuomnilemdc.fo	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
2		fof 5450	. . . . . 6 ⊢ (𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
4	3	ffvelcdmda 5664	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (𝐹‘𝑋) ∈ (𝐴 ⊔ 𝐵))
5		djur 7081	. . . 4 ⊢ ((𝐹‘𝑋) ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)))
6	4, 5	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)))
7		nfv 1538	. . . . . . . 8 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑋 ∈ 𝑂)
8		nfre1 2530	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)
9	7, 8	nfan 1575	. . . . . . 7 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧))
10		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧))
11		fveq2 5527	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
12	11	eqeq2d 2199	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑋) = (inr‘𝑧) ↔ (𝐹‘𝑋) = (inr‘𝑤)))
13	12	cbvrexv 2716	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧) ↔ ∃𝑤 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑤))
14	10, 13	sylib 122	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∃𝑤 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑤))
15		vex 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
16		vex 2752	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
17		djune 7090	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (inl‘𝑧) ≠ (inr‘𝑤)
19		neeq2 2371	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → ((inl‘𝑧) ≠ (𝐹‘𝑋) ↔ (inl‘𝑧) ≠ (inr‘𝑤)))
20	18, 19	mpbiri 168	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → (inl‘𝑧) ≠ (𝐹‘𝑋))
21	20	necomd 2443	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → (𝐹‘𝑋) ≠ (inl‘𝑧))
22	21	neneqd 2378	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → ¬ (𝐹‘𝑋) = (inl‘𝑧))
23	22	a1i 9	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ((𝐹‘𝑋) = (inr‘𝑤) → ¬ (𝐹‘𝑋) = (inl‘𝑧)))
24	23	rexlimdvw 2608	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → (∃𝑤 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑤) → ¬ (𝐹‘𝑋) = (inl‘𝑧)))
25	14, 24	mpd 13	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ¬ (𝐹‘𝑋) = (inl‘𝑧))
26	25	a1d 22	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → (𝑧 ∈ 𝐴 → ¬ (𝐹‘𝑋) = (inl‘𝑧)))
27	9, 26	ralrimi 2558	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∀𝑧 ∈ 𝐴 ¬ (𝐹‘𝑋) = (inl‘𝑧))
28		ralnex 2475	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 ¬ (𝐹‘𝑋) = (inl‘𝑧) ↔ ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))
29	27, 28	sylib 122	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))
30	29	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧) → ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧)))
31	30	orim2d 789	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → ((∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))))
32	6, 31	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧)))
33		df-dc 836	. 2 ⊢ (DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ↔ (∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧) ∨ ¬ ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧)))
34	32, 33	sylibr 134	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1363   ∈ wcel 2158   ≠ wne 2357  ∀wral 2465  ∃wrex 2466  Vcvv 2749  ⟶wf 5224  –onto→wfo 5226  ‘cfv 5228   ⊔ cdju 7049  inlcinl 7057  inrcinr 7058

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050  df-inl 7059  df-inr 7060

This theorem is referenced by:  fodjuf  7156  fodjum  7157  fodju0  7158