Theorem fodjuomnilemres 7439

Description: Lemma for fodjuomni 7440. 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 5669	. . . . . 6 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
2	1	eqeq1d 2241	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
3	2	rexbidv 2543	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅))
4	1	eqeq1d 2241	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
5	4	ralbidv 2542	. . . 4 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
6	3, 5	orbi12d 801	. . 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 7428	. . . . 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 147	. . 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 7436	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
14	6, 10, 13	rspcdva 2926	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
15	11	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
16		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)
17		fveqeq2 5679	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
18	17	cbvrexv 2779	. . . . . 6 ⊢ (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ↔ ∃𝑣 ∈ 𝑂 (𝑃‘𝑣) = ∅)
19	16, 18	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑣 ∈ 𝑂 (𝑃‘𝑣) = ∅)
20	15, 12, 19	fodjum 7437	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅) → ∃𝑥 𝑥 ∈ 𝐴)
21	20	ex 115	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ → ∃𝑥 𝑥 ∈ 𝐴))
22	11	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))
23		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)
24	22, 12, 23	fodju0 7438	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
25	24	ex 115	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
26	21, 25	orim12d 794	. 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 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  ∅c0 3508  ifcif 3620   ↦ cmpt 4171  –onto→wfo 5350  ‘cfv 5352  (class class class)co 6050  1_oc1o 6640  2_oc2o 6641   ↑_𝑚 cmap 6882   ⊔ cdju 7328  inlcinl 7336  Omnicomni 7425

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-map 6884  df-dju 7329  df-inl 7338  df-inr 7339  df-omni 7426

This theorem is referenced by:  fodjuomni  7440