Theorem fodjumkvlemres 7261

Description: Lemma for fodjumkv 7262. 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 7249	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
6	5	ex 115	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
7	6	necon3ad 2418	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
8		fveq1 5575	. . . . . . 7 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
9	8	eqeq1d 2214	. . . . . 6 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
10	9	ralbidv 2506	. . . . 5 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
11	10	notbid 669	. . . 4 ⊢ (𝑓 = 𝑃 → (¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
12	8	eqeq1d 2214	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
13	12	rexbidv 2507	. . . 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 7256	. . . . 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 7247	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑀))
20	14, 18, 19	rspcdva 2882	. 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 5585	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
24	23	cbvrexv 2739	. . . . 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 7248	. . 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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1515   ∈ wcel 2176   ≠ wne 2376  ∀wral 2484  ∃wrex 2485  ∅c0 3460  ifcif 3571   ↦ cmpt 4105  –onto→wfo 5269  ‘cfv 5271  (class class class)co 5944  1_oc1o 6495  2_oc2o 6496   ↑_𝑚 cmap 6735   ⊔ cdju 7139  inlcinl 7147  Markovcmarkov 7253

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-1o 6502  df-2o 6503  df-map 6737  df-dju 7140  df-inl 7149  df-inr 7150  df-markov 7254

This theorem is referenced by:  fodjumkv  7262