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Mirrors > Home > ILE Home > Th. List > fodjuf | GIF version |
Description: Lemma for fodjuomni 6933 and fodjumkv 6945. 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 6268 | . . . . 5 ⊢ ∅ ∈ 2o | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → ∅ ∈ 2o) |
3 | 1lt2o 6269 | . . . . 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 6928 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧)) |
7 | 2, 4, 6 | ifcldcd 3454 | . . 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 5506 | . 2 ⊢ (𝜑 → 𝑃:𝑂⟶2o) |
10 | 2onn 6347 | . . . 4 ⊢ 2o ∈ ω | |
11 | 10 | a1i 9 | . . 3 ⊢ (𝜑 → 2o ∈ ω) |
12 | fodjuf.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
13 | 11, 12 | elmapd 6486 | . 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 1299 ∈ wcel 1448 ∃wrex 2376 ∅c0 3310 ifcif 3421 ↦ cmpt 3929 ωcom 4442 ⟶wf 5055 –onto→wfo 5057 ‘cfv 5059 (class class class)co 5706 1oc1o 6236 2oc2o 6237 ↑𝑚 cmap 6472 ⊔ cdju 6837 inlcinl 6845 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-1o 6243 df-2o 6244 df-map 6474 df-dju 6838 df-inl 6847 df-inr 6848 |
This theorem is referenced by: fodjuomnilemres 6932 fodjumkvlemres 6944 |
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