Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fodjumkv | GIF version |
Description: A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
fodjumkv.o | ⊢ (𝜑 → 𝑀 ∈ Markov) |
fodjumkv.fo | ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵)) |
Ref | Expression |
---|---|
fodjumkv | ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjumkv.o | . 2 ⊢ (𝜑 → 𝑀 ∈ Markov) | |
2 | fodjumkv.fo | . 2 ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵)) | |
3 | fveq2 5389 | . . . . . . 7 ⊢ (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧)) | |
4 | 3 | eqeq2d 2129 | . . . . . 6 ⊢ (𝑏 = 𝑧 → ((𝐹‘𝑎) = (inl‘𝑏) ↔ (𝐹‘𝑎) = (inl‘𝑧))) |
5 | 4 | cbvrexv 2632 | . . . . 5 ⊢ (∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧)) |
6 | ifbi 3462 | . . . . 5 ⊢ ((∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧)) → if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) | |
7 | 5, 6 | ax-mp 5 | . . . 4 ⊢ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) |
8 | 7 | mpteq2i 3985 | . . 3 ⊢ (𝑎 ∈ 𝑀 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) |
9 | fveqeq2 5398 | . . . . . 6 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑎) = (inl‘𝑧) ↔ (𝐹‘𝑦) = (inl‘𝑧))) | |
10 | 9 | rexbidv 2415 | . . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧))) |
11 | 10 | ifbid 3463 | . . . 4 ⊢ (𝑎 = 𝑦 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
12 | 11 | cbvmptv 3994 | . . 3 ⊢ (𝑎 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
13 | 8, 12 | eqtri 2138 | . 2 ⊢ (𝑎 ∈ 𝑀 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
14 | 1, 2, 13 | fodjumkvlemres 7001 | 1 ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ≠ wne 2285 ∃wrex 2394 ∅c0 3333 ifcif 3444 ↦ cmpt 3959 –onto→wfo 5091 ‘cfv 5093 1oc1o 6274 ⊔ cdju 6890 inlcinl 6898 Markovcmarkov 6993 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-1o 6281 df-2o 6282 df-map 6512 df-dju 6891 df-inl 6900 df-inr 6901 df-markov 6994 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |