Theorem fodjumkvlemres 7499

Description: Lemma for fodjumkv 7500. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)

Hypotheses

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

Assertion

Ref	Expression
fodjumkvlemres	⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))

Distinct variable groups:   𝜑,𝑦,𝑧   𝑦,𝑀,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)   𝑀(𝑥)

Proof of Theorem fodjumkvlemres

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

Step	Hyp	Ref	Expression
1		fodjumkv.fo	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))
2	1	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))
3		fodjumkv.p	. . . . 5 ⊢ 𝑃 = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))
4		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o)
5	2, 3, 4	fodju0 7487	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
6	5	ex 115	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
7	6	necon3ad 2462	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
8		fveq1 5694	. . . . . . 7 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
9	8	eqeq1d 2247	. . . . . 6 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
10	9	ralbidv 2550	. . . . 5 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
11	10	notbid 677	. . . 4 ⊢ (𝑓 = 𝑃 → (¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
12	8	eqeq1d 2247	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
13	12	rexbidv 2551	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅))
14	11, 13	imbi12d 234	. . 3 ⊢ (𝑓 = 𝑃 → ((¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅) ↔ (¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅)))
15		fodjumkv.o	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Markov)
16		ismkvmap 7494	. . . . 5 ⊢ (𝑀 ∈ Markov → (𝑀 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝑀)(¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅)))
17	16	ibi 176	. . . 4 ⊢ (𝑀 ∈ Markov → ∀𝑓 ∈ (2o ↑𝑚 𝑀)(¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅))
18	15, 17	syl 14	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 𝑀)(¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅))
19	1, 3, 15	fodjuf 7485	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑀))
20	14, 18, 19	rspcdva 2934	. 2 ⊢ (𝜑 → (¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅))
21	1	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))
22		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅)
23		fveqeq2 5704	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
24	23	cbvrexv 2787	. . . . 5 ⊢ (∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅ ↔ ∃𝑣 ∈ 𝑀 (𝑃‘𝑣) = ∅)
25	22, 24	sylib 122	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑣 ∈ 𝑀 (𝑃‘𝑣) = ∅)
26	21, 3, 25	fodjum 7486	. . 3 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑥 𝑥 ∈ 𝐴)
27	26	ex 115	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅ → ∃𝑥 𝑥 ∈ 𝐴))
28	7, 20, 27	3syld 57	1 ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529  ∅c0 3520  ifcif 3638   ↦ cmpt 4192  –onto→wfo 5375  ‘cfv 5377  (class class class)co 6085  1_oc1o 6680  2_oc2o 6681   ↑_𝑚 cmap 6922   ⊔ cdju 7377  inlcinl 7385  Markovcmarkov 7491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-map 6924  df-dju 7378  df-inl 7387  df-inr 7388  df-markov 7492

This theorem is used by:  fodjumkv  7500