Theorem fodjuomnilemres 6891

Description: Lemma for fodjuomni 6892. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)

Hypotheses

Ref	Expression
fodjuomni.o	⊢ (𝜑 → 𝑂 ∈ Omni)
fodjuomni.fo	⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
fodjuomni.p	⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))

Assertion

Ref	Expression
fodjuomnilemres	⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))

Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑂,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹   𝑥,𝐴,𝑧   𝑦,𝐴   𝑦,𝐹   𝑦,𝑃,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑦)   𝑃(𝑥)   𝐹(𝑥)   𝑂(𝑥)

Proof of Theorem fodjuomnilemres

Dummy variables 𝑣 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5339	. . . . . 6 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
2	1	eqeq1d 2103	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
3	2	rexbidv 2392	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅))
4	1	eqeq1d 2103	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
5	4	ralbidv 2391	. . . 4 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
6	3, 5	orbi12d 745	. . 3 ⊢ (𝑓 = 𝑃 → ((∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o) ↔ (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)))
7		fodjuomni.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ Omni)
8		isomnimap 6880	. . . . 5 ⊢ (𝑂 ∈ Omni → (𝑂 ∈ Omni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝑂)(∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o)))
9	7, 8	syl 14	. . . 4 ⊢ (𝜑 → (𝑂 ∈ Omni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝑂)(∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o)))
10	7, 9	mpbid 146	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 𝑂)(∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o))
11		fodjuomni.fo	. . . 4 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
12		fodjuomni.p	. . . 4 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
13	11, 12, 7	fodjuf 6888	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
14	6, 10, 13	rspcdva 2741	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
15	11	adantr 271	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
16		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)
17		fveqeq2 5349	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
18	17	cbvrexv 2605	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ↔ ∃𝑣 ∈ 𝑂 (𝑃‘𝑣) = ∅)
19	16, 18	sylib 121	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑣 ∈ 𝑂 (𝑃‘𝑣) = ∅)
20	15, 12, 19	fodjum 6889	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑥 𝑥 ∈ 𝐴)
21	20	ex 114	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ → ∃𝑥 𝑥 ∈ 𝐴))
22	11	adantr 271	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
23		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
24	22, 12, 23	fodju0 6890	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
25	24	ex 114	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
26	21, 25	orim12d 738	. 2 ⊢ (𝜑 → ((∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅)))
27	14, 26	mpd 13	1 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  ∅c0 3302  ifcif 3413   ↦ cmpt 3921  –onto→wfo 5047  ‘cfv 5049  (class class class)co 5690  1_oc1o 6212  2_oc2o 6213   ↑_𝑚 cmap 6445   ⊔ cdju 6810  inlcinl 6817  Omnicomni 6875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-1o 6219  df-2o 6220  df-map 6447  df-dju 6811  df-inl 6819  df-inr 6820  df-omni 6877

This theorem is referenced by:  fodjuomni  6892