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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 5375 | . . . . . . 7 ⊢ (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧)) | |
4 | 3 | eqeq2d 2126 | . . . . . 6 ⊢ (𝑏 = 𝑧 → ((𝐹‘𝑎) = (inl‘𝑏) ↔ (𝐹‘𝑎) = (inl‘𝑧))) |
5 | 4 | cbvrexv 2629 | . . . . 5 ⊢ (∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧)) |
6 | ifbi 3458 | . . . . 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 7 | . . . 4 ⊢ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) |
8 | 7 | mpteq2i 3975 | . . 3 ⊢ (𝑎 ∈ 𝑀 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) |
9 | fveqeq2 5384 | . . . . . 6 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑎) = (inl‘𝑧) ↔ (𝐹‘𝑦) = (inl‘𝑧))) | |
10 | 9 | rexbidv 2412 | . . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧))) |
11 | 10 | ifbid 3459 | . . . 4 ⊢ (𝑎 = 𝑦 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
12 | 11 | cbvmptv 3984 | . . 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 2135 | . 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 6983 | 1 ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1314 ∃wex 1451 ∈ wcel 1463 ≠ wne 2282 ∃wrex 2391 ∅c0 3329 ifcif 3440 ↦ cmpt 3949 –onto→wfo 5079 ‘cfv 5081 1oc1o 6260 ⊔ cdju 6874 inlcinl 6882 Markovcmarkov 6975 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-1o 6267 df-2o 6268 df-map 6498 df-dju 6875 df-inl 6884 df-inr 6885 df-markov 6976 |
This theorem is referenced by: (None) |
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