Theorem fodjuomnilemres 7311

Description: Lemma for fodjuomni 7312. 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 5625	. . . . . 6 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
2	1	eqeq1d 2238	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
3	2	rexbidv 2531	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑂 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅))
4	1	eqeq1d 2238	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
5	4	ralbidv 2530	. . . 4 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑂 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o))
6	3, 5	orbi12d 798	. . 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 7300	. . . . 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 7308	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
14	6, 10, 13	rspcdva 2912	. 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 5635	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
18	17	cbvrexv 2766	. . . . . 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 7309	. . . 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 7310	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
25	24	ex 115	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
26	21, 25	orim12d 791	. 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 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∅c0 3491  ifcif 3602   ↦ cmpt 4144  –onto→wfo 5315  ‘cfv 5317  (class class class)co 6000  1_oc1o 6553  2_oc2o 6554   ↑_𝑚 cmap 6793   ⊔ cdju 7200  inlcinl 7208  Omnicomni 7297

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  ax-setind 4628

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-rab 2517  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-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  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-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  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-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-1o 6560  df-2o 6561  df-map 6795  df-dju 7201  df-inl 7210  df-inr 7211  df-omni 7298

This theorem is referenced by:  fodjuomni  7312