Theorem fodjuomnilemdc 7260

Description: Lemma for fodjuomni 7265. 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 5509	. . . . . 6 ⊢ (𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
4	3	ffvelcdmda 5727	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (𝐹‘𝑋) ∈ (𝐴 ⊔ 𝐵))
5		djur 7185	. . . 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 1552	. . . . . . . 8 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑋 ∈ 𝑂)
8		nfre1 2550	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)
9	7, 8	nfan 1589	. . . . . . 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 5588	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
12	11	eqeq2d 2218	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑋) = (inr‘𝑧) ↔ (𝐹‘𝑋) = (inr‘𝑤)))
13	12	cbvrexv 2740	. . . . . . . . . 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 2776	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
16		vex 2776	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
17		djune 7194	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (inl‘𝑧) ≠ (inr‘𝑤)
19		neeq2 2391	. . . . . . . . . . . . . 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 2463	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → (𝐹‘𝑋) ≠ (inl‘𝑧))
22	21	neneqd 2398	. . . . . . . . . . 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 2628	. . . . . . . . 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 2578	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∀𝑧 ∈ 𝐴 ¬ (𝐹‘𝑋) = (inl‘𝑧))
28		ralnex 2495	. . . . . 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 790	. . 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 837	. 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 710  DECID wdc 836   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∀wral 2485  ∃wrex 2486  Vcvv 2773  ⟶wf 5275  –onto→wfo 5277  ‘cfv 5279   ⊔ cdju 7153  inlcinl 7161  inrcinr 7162

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-1st 6238  df-2nd 6239  df-1o 6514  df-dju 7154  df-inl 7163  df-inr 7164

This theorem is referenced by:  fodjuf  7261  fodjum  7262  fodju0  7263