Theorem fodjumkvlemres 7135

Description: Lemma for fodjumkv 7136. 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 274	. . . . 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 109	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o)
5	2, 3, 4	fodju0 7123	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
6	5	ex 114	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
7	6	necon3ad 2382	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
8		fveq1 5495	. . . . . . 7 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
9	8	eqeq1d 2179	. . . . . 6 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
10	9	ralbidv 2470	. . . . 5 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
11	10	notbid 662	. . . 4 ⊢ (𝑓 = 𝑃 → (¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
12	8	eqeq1d 2179	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
13	12	rexbidv 2471	. . . 4 ⊢ (𝑓 = 𝑃 → (∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅ ↔ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅))
14	11, 13	imbi12d 233	. . 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 7130	. . . . 5 ⊢ (𝑀 ∈ Markov → (𝑀 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝑀)(¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑓‘𝑤) = ∅)))
17	16	ibi 175	. . . 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 7121	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑀))
20	14, 18, 19	rspcdva 2839	. 2 ⊢ (𝜑 → (¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅))
21	1	adantr 274	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))
22		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅)
23		fveqeq2 5505	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
24	23	cbvrexv 2697	. . . . 5 ⊢ (∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅ ↔ ∃𝑣 ∈ 𝑀 (𝑃‘𝑣) = ∅)
25	22, 24	sylib 121	. . . 4 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑣 ∈ 𝑀 (𝑃‘𝑣) = ∅)
26	21, 3, 25	fodjum 7122	. . 3 ⊢ ((𝜑 ∧ ∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅) → ∃𝑥 𝑥 ∈ 𝐴)
27	26	ex 114	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑀 (𝑃‘𝑤) = ∅ → ∃𝑥 𝑥 ∈ 𝐴))
28	7, 20, 27	3syld 57	1 ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449  ∅c0 3414  ifcif 3526   ↦ cmpt 4050  –onto→wfo 5196  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  2_oc2o 6389   ↑_𝑚 cmap 6626   ⊔ cdju 7014  inlcinl 7022  Markovcmarkov 7127

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-1o 6395  df-2o 6396  df-map 6628  df-dju 7015  df-inl 7024  df-inr 7025  df-markov 7128

This theorem is referenced by:  fodjumkv  7136