Theorem fodjuomnilemdc 7162

Description: Lemma for fodjuomni 7167. 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 5454	. . . . . 6 ⊢ (𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
4	3	ffvelcdmda 5668	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (𝐹‘𝑋) ∈ (𝐴 ⊔ 𝐵))
5		djur 7088	. . . 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 1539	. . . . . . . 8 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑋 ∈ 𝑂)
8		nfre1 2533	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)
9	7, 8	nfan 1576	. . . . . . 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 5531	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
12	11	eqeq2d 2201	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑋) = (inr‘𝑧) ↔ (𝐹‘𝑋) = (inr‘𝑤)))
13	12	cbvrexv 2719	. . . . . . . . . 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 2755	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
16		vex 2755	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
17		djune 7097	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (inl‘𝑧) ≠ (inr‘𝑤)
19		neeq2 2374	. . . . . . . . . . . . . 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 2446	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → (𝐹‘𝑋) ≠ (inl‘𝑧))
22	21	neneqd 2381	. . . . . . . . . . 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 2611	. . . . . . . . 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 2561	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∀𝑧 ∈ 𝐴 ¬ (𝐹‘𝑋) = (inl‘𝑧))
28		ralnex 2478	. . . . . 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 1364   ∈ wcel 2160   ≠ wne 2360  ∀wral 2468  ∃wrex 2469  Vcvv 2752  ⟶wf 5228  –onto→wfo 5230  ‘cfv 5232   ⊔ cdju 7056  inlcinl 7064  inrcinr 7065

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 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-1st 6160  df-2nd 6161  df-1o 6436  df-dju 7057  df-inl 7066  df-inr 7067

This theorem is referenced by:  fodjuf  7163  fodjum  7164  fodju0  7165