Theorem fodjuomnilemres 7028

Description: Lemma for fodjuomni 7029. 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 5428	. . . . . 6 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
2	1	eqeq1d 2149	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
3	2	rexbidv 2439	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅))
4	1	eqeq1d 2149	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
5	4	ralbidv 2438	. . . 4 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
6	3, 5	orbi12d 783	. . 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 7017	. . . . 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 7025	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
14	6, 10, 13	rspcdva 2798	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅ ∨ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
15	11	adantr 274	. . . . 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 5438	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
18	17	cbvrexv 2658	. . . . . 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 7026	. . . 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 274	. . . . 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 7027	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
25	24	ex 114	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
26	21, 25	orim12d 776	. 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 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∅c0 3368  ifcif 3479   ↦ cmpt 3997  –onto→wfo 5129  ‘cfv 5131  (class class class)co 5782  1_oc1o 6314  2_oc2o 6315   ↑_𝑚 cmap 6550   ⊔ cdju 6930  inlcinl 6938  Omnicomni 7012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-1o 6321  df-2o 6322  df-map 6552  df-dju 6931  df-inl 6940  df-inr 6941  df-omni 7014

This theorem is referenced by:  fodjuomni  7029