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Mirrors > Home > ILE Home > Th. List > fodjumkvlemres | GIF version |
Description: Lemma for fodjumkv 7116. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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)) |
Ref | Expression |
---|---|
fodjumkvlemres | ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) |
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 7103 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o) → 𝐴 = ∅) |
6 | 5 | ex 114 | . . 3 ⊢ (𝜑 → (∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o → 𝐴 = ∅)) |
7 | 6 | necon3ad 2376 | . 2 ⊢ (𝜑 → (𝐴 ≠ ∅ → ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o)) |
8 | fveq1 5480 | . . . . . . 7 ⊢ (𝑓 = 𝑃 → (𝑓‘𝑤) = (𝑃‘𝑤)) | |
9 | 8 | eqeq1d 2173 | . . . . . 6 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = 1o ↔ (𝑃‘𝑤) = 1o)) |
10 | 9 | ralbidv 2464 | . . . . 5 ⊢ (𝑓 = 𝑃 → (∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o)) |
11 | 10 | notbid 657 | . . . 4 ⊢ (𝑓 = 𝑃 → (¬ ∀𝑤 ∈ 𝑀 (𝑓‘𝑤) = 1o ↔ ¬ ∀𝑤 ∈ 𝑀 (𝑃‘𝑤) = 1o)) |
12 | 8 | eqeq1d 2173 | . . . . 5 ⊢ (𝑓 = 𝑃 → ((𝑓‘𝑤) = ∅ ↔ (𝑃‘𝑤) = ∅)) |
13 | 12 | rexbidv 2465 | . . . 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 7110 | . . . . 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 7101 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑀)) |
20 | 14, 18, 19 | rspcdva 2831 | . 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 5490 | . . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑣) = ∅)) | |
24 | 23 | cbvrexv 2691 | . . . . 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 7102 | . . 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)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ≠ wne 2334 ∀wral 2442 ∃wrex 2443 ∅c0 3405 ifcif 3516 ↦ cmpt 4038 –onto→wfo 5181 ‘cfv 5183 (class class class)co 5837 1oc1o 6369 2oc2o 6370 ↑𝑚 cmap 6606 ⊔ cdju 6994 inlcinl 7002 Markovcmarkov 7107 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-1o 6376 df-2o 6377 df-map 6608 df-dju 6995 df-inl 7004 df-inr 7005 df-markov 7108 |
This theorem is referenced by: fodjumkv 7116 |
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