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Mirrors > Home > ILE Home > Th. List > fodjuf | GIF version |
Description: Lemma for fodjuomni 7021 and fodjumkv 7034. Domain and range of 𝑃. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
fodjuf.fo | ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) |
fodjuf.p | ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) |
fodjuf.o | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
Ref | Expression |
---|---|
fodjuf | ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt2o 6338 | . . . . 5 ⊢ ∅ ∈ 2o | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → ∅ ∈ 2o) |
3 | 1lt2o 6339 | . . . . 5 ⊢ 1o ∈ 2o | |
4 | 3 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → 1o ∈ 2o) |
5 | fodjuf.fo | . . . . 5 ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵)) | |
6 | 5 | fodjuomnilemdc 7016 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧)) |
7 | 2, 4, 6 | ifcldcd 3507 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o) ∈ 2o) |
8 | fodjuf.p | . . 3 ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o)) | |
9 | 7, 8 | fmptd 5574 | . 2 ⊢ (𝜑 → 𝑃:𝑂⟶2o) |
10 | 2onn 6417 | . . . 4 ⊢ 2o ∈ ω | |
11 | 10 | a1i 9 | . . 3 ⊢ (𝜑 → 2o ∈ ω) |
12 | fodjuf.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
13 | 11, 12 | elmapd 6556 | . 2 ⊢ (𝜑 → (𝑃 ∈ (2o ↑𝑚 𝑂) ↔ 𝑃:𝑂⟶2o)) |
14 | 9, 13 | mpbird 166 | 1 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ∅c0 3363 ifcif 3474 ↦ cmpt 3989 ωcom 4504 ⟶wf 5119 –onto→wfo 5121 ‘cfv 5123 (class class class)co 5774 1oc1o 6306 2oc2o 6307 ↑𝑚 cmap 6542 ⊔ cdju 6922 inlcinl 6930 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-1o 6313 df-2o 6314 df-map 6544 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: fodjuomnilemres 7020 fodjumkvlemres 7033 |
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