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Mirrors > Home > ILE Home > Th. List > elmpocl | GIF version |
Description: If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
elmpocl.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
elmpocl | ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmpocl.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | df-mpo 5841 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2185 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 4799 | . . . 4 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 5915 | . . . 4 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵) | |
6 | 4, 5 | eqsstri 3169 | . . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵) |
7 | 1 | mpofun 5935 | . . . . . 6 ⊢ Fun 𝐹 |
8 | funrel 5199 | . . . . . 6 ⊢ (Fun 𝐹 → Rel 𝐹) | |
9 | 7, 8 | ax-mp 5 | . . . . 5 ⊢ Rel 𝐹 |
10 | relelfvdm 5512 | . . . . 5 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉)) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) | |
11 | 9, 10 | mpan 421 | . . . 4 ⊢ (𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
12 | df-ov 5839 | . . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘〈𝑆, 𝑇〉) | |
13 | 11, 12 | eleq2s 2259 | . . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
14 | 6, 13 | sseldi 3135 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵)) |
15 | opelxp 4628 | . 2 ⊢ (〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) | |
16 | 14, 15 | sylib 121 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 〈cop 3573 × cxp 4596 dom cdm 4598 Rel wrel 4603 Fun wfun 5176 ‘cfv 5182 (class class class)co 5836 {coprab 5837 ∈ cmpo 5838 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 |
This theorem is referenced by: elmpocl1 6031 elmpocl2 6032 elovmpo 6033 elpmi 6624 elmapex 6626 pmsspw 6640 ixxssxr 9827 elixx3g 9828 ixxssixx 9829 eliooxr 9854 elfz2 9942 restsspw 12502 restrcl 12708 ssrest 12723 iscn2 12741 ishmeo 12845 limcrcl 13168 |
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