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Mirrors > Home > ILE Home > Th. List > elmpt2cl | GIF version |
Description: If a two-parameter class is not empty, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
elmpt2cl.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
elmpt2cl | ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmpt2cl.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | df-mpt2 5595 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2103 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 4594 | . . . 4 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 5664 | . . . 4 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵) | |
6 | 4, 5 | eqsstri 3040 | . . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵) |
7 | 1 | mpt2fun 5681 | . . . . . 6 ⊢ Fun 𝐹 |
8 | funrel 4985 | . . . . . 6 ⊢ (Fun 𝐹 → Rel 𝐹) | |
9 | 7, 8 | ax-mp 7 | . . . . 5 ⊢ Rel 𝐹 |
10 | relelfvdm 5280 | . . . . 5 ⊢ ((Rel 𝐹 ∧ 𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉)) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) | |
11 | 9, 10 | mpan 415 | . . . 4 ⊢ (𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
12 | df-ov 5593 | . . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘〈𝑆, 𝑇〉) | |
13 | 11, 12 | eleq2s 2177 | . . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
14 | 6, 13 | sseldi 3008 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵)) |
15 | opelxp 4429 | . 2 ⊢ (〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) | |
16 | 14, 15 | sylib 120 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 〈cop 3425 × cxp 4398 dom cdm 4400 Rel wrel 4405 Fun wfun 4962 ‘cfv 4968 (class class class)co 5590 {coprab 5591 ↦ cmpt2 5592 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-ov 5593 df-oprab 5594 df-mpt2 5595 |
This theorem is referenced by: elmpt2cl1 5777 elmpt2cl2 5778 elovmpt2 5779 elpmi 6353 elmapex 6355 pmsspw 6369 ixxssxr 9212 elixx3g 9213 ixxssixx 9214 eliooxr 9239 elfz2 9325 |
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