Theorem fodjumkvlemres 7175

Description: Lemma for fodjumkv 7176. 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 7163	. . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐴 = ∅)
6	5	ex 115	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → 𝐴 = ∅))
7	6	necon3ad 2402	. 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
8		fveq1 5529	. . . . . . 7 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤))
9	8	eqeq1d 2198	. . . . . 6 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o))
10	9	ralbidv 2490	. . . . 5 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
11	10	notbid 668	. . . 4 ⊢ (𝑓 = 𝑃 → (¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o))
12	8	eqeq1d 2198	. . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅))
13	12	rexbidv 2491	. . . 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 7170	. . . . 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 7161	. . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑀))
20	14, 18, 19	rspcdva 2861	. 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 5539	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅))
24	23	cbvrexv 2719	. . . . 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 7162	. . 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 1364  ∃wex 1503   ∈ wcel 2160   ≠ wne 2360  ∀wral 2468  ∃wrex 2469  ∅c0 3437  ifcif 3549   ↦ cmpt 4079  –onto→wfo 5229  ‘cfv 5231  (class class class)co 5891  1_oc1o 6428  2_oc2o 6429   ↑_𝑚 cmap 6666   ⊔ cdju 7054  inlcinl 7062  Markovcmarkov 7167

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-1o 6435  df-2o 6436  df-map 6668  df-dju 7055  df-inl 7064  df-inr 7065  df-markov 7168

This theorem is referenced by:  fodjumkv  7176