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Mirrors > Home > ILE Home > Th. List > fodjuomni | GIF version |
Description: A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.) |
Ref | Expression |
---|---|
fodjuomni.o | ⊢ (𝜑 → 𝑂 ∈ Omni) |
fodjuomni.fo | ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) |
Ref | Expression |
---|---|
fodjuomni | ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjuomni.o | . 2 ⊢ (𝜑 → 𝑂 ∈ Omni) | |
2 | fodjuomni.fo | . 2 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) | |
3 | fveq2 5414 | . . . . . . 7 ⊢ (𝑏 = 𝑧 → (inl‘𝑏) = (inl‘𝑧)) | |
4 | 3 | eqeq2d 2149 | . . . . . 6 ⊢ (𝑏 = 𝑧 → ((𝐹‘𝑎) = (inl‘𝑏) ↔ (𝐹‘𝑎) = (inl‘𝑧))) |
5 | 4 | cbvrexv 2653 | . . . . 5 ⊢ (∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧)) |
6 | ifbi 3487 | . . . . 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 4010 | . . 3 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑎 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) |
9 | fveq2 5414 | . . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦)) | |
10 | 9 | eqeq1d 2146 | . . . . . 6 ⊢ (𝑎 = 𝑦 → ((𝐹‘𝑎) = (inl‘𝑧) ↔ (𝐹‘𝑦) = (inl‘𝑧))) |
11 | 10 | rexbidv 2436 | . . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧) ↔ ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧))) |
12 | 11 | ifbid 3488 | . . . 4 ⊢ (𝑎 = 𝑦 → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o) = if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
13 | 12 | cbvmptv 4019 | . . 3 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑧), ∅, 1o)) = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
14 | 8, 13 | eqtri 2158 | . 2 ⊢ (𝑎 ∈ 𝑂 ↦ if(∃𝑏 ∈ 𝐴 (𝐹‘𝑎) = (inl‘𝑏), ∅, 1o)) = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
15 | 1, 2, 14 | fodjuomnilemres 7013 | 1 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃wrex 2415 ∅c0 3358 ifcif 3469 ↦ cmpt 3984 –onto→wfo 5116 ‘cfv 5118 1oc1o 6299 ⊔ cdju 6915 inlcinl 6923 Omnicomni 6997 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-1o 6306 df-2o 6307 df-map 6537 df-dju 6916 df-inl 6925 df-inr 6926 df-omni 6999 |
This theorem is referenced by: ctssexmid 7017 exmidunben 11928 exmidsbthrlem 13206 sbthomlem 13209 |
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