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Mirrors > Home > ILE Home > Th. List > fodjuf | GIF version |
Description: Lemma for fodjuomni 7165 and fodjumkv 7176. 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 6460 | . . . . 5 ⊢ ∅ ∈ 2o | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → ∅ ∈ 2o) |
3 | 1lt2o 6461 | . . . . 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 7160 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧)) |
7 | 2, 4, 6 | ifcldcd 3585 | . . 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 5686 | . 2 ⊢ (𝜑 → 𝑃:𝑂⟶2o) |
10 | 2onn 6540 | . . . 4 ⊢ 2o ∈ ω | |
11 | 10 | a1i 9 | . . 3 ⊢ (𝜑 → 2o ∈ ω) |
12 | fodjuf.o | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
13 | 11, 12 | elmapd 6680 | . 2 ⊢ (𝜑 → (𝑃 ∈ (2o ↑𝑚 𝑂) ↔ 𝑃:𝑂⟶2o)) |
14 | 9, 13 | mpbird 167 | 1 ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 ∅c0 3437 ifcif 3549 ↦ cmpt 4079 ωcom 4604 ⟶wf 5227 –onto→wfo 5229 ‘cfv 5231 (class class class)co 5891 1oc1o 6428 2oc2o 6429 ↑𝑚 cmap 6666 ⊔ cdju 7054 inlcinl 7062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-1o 6435 df-2o 6436 df-map 6668 df-dju 7055 df-inl 7064 df-inr 7065 |
This theorem is referenced by: fodjuomnilemres 7164 fodjumkvlemres 7175 |
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