Theorem fodjuomnilemdc 7307

Description: Lemma for fodjuomni 7312. 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 5547	. . . . . 6 ⊢ (𝐹:𝑂–onto→(𝐴 ⊔ 𝐵) → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐹:𝑂⟶(𝐴 ⊔ 𝐵))
4	3	ffvelcdmda 5769	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → (𝐹‘𝑋) ∈ (𝐴 ⊔ 𝐵))
5		djur 7232	. . . 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 1574	. . . . . . . 8 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑋 ∈ 𝑂)
8		nfre1 2573	. . . . . . . 8 ⊢ Ⅎ𝑧∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)
9	7, 8	nfan 1611	. . . . . . 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 5626	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (inr‘𝑧) = (inr‘𝑤))
12	11	eqeq2d 2241	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑋) = (inr‘𝑧) ↔ (𝐹‘𝑋) = (inr‘𝑤)))
13	12	cbvrexv 2766	. . . . . . . . . 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 2802	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
16		vex 2802	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
17		djune 7241	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ V ∧ 𝑤 ∈ V) → (inl‘𝑧) ≠ (inr‘𝑤))
18	15, 16, 17	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (inl‘𝑧) ≠ (inr‘𝑤)
19		neeq2 2414	. . . . . . . . . . . . . 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 2486	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑋) = (inr‘𝑤) → (𝐹‘𝑋) ≠ (inl‘𝑧))
22	21	neneqd 2421	. . . . . . . . . . 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 2652	. . . . . . . . 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 2601	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑂) ∧ ∃𝑧 ∈ 𝐵 (𝐹‘𝑋) = (inr‘𝑧)) → ∀𝑧 ∈ 𝐴 ¬ (𝐹‘𝑋) = (inl‘𝑧))
28		ralnex 2518	. . . . . 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 793	. . 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 840	. 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 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  ∃wrex 2509  Vcvv 2799  ⟶wf 5313  –onto→wfo 5315  ‘cfv 5317   ⊔ cdju 7200  inlcinl 7208  inrcinr 7209

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-1o 6560  df-dju 7201  df-inl 7210  df-inr 7211

This theorem is referenced by:  fodjuf  7308  fodjum  7309  fodju0  7310