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Mirrors > Home > MPE Home > Th. List > elmpt2cl | Structured version Visualization version 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 6554 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2632 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5247 | . . . 4 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 6640 | . . . 4 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵) | |
6 | 4, 5 | eqsstri 3598 | . . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵) |
7 | elfvdm 6130 | . . . 4 ⊢ (𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) | |
8 | df-ov 6552 | . . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘〈𝑆, 𝑇〉) | |
9 | 7, 8 | eleq2s 2706 | . . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
10 | 6, 9 | sseldi 3566 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵)) |
11 | opelxp 5070 | . 2 ⊢ (〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) | |
12 | 10, 11 | sylib 207 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 {coprab 6550 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: elmpt2cl1 6775 elmpt2cl2 6776 elovmpt2 6777 elovmpt2rab 6778 elovmpt2rab1 6779 el2mpt2csbcl 7137 ixxssixx 12060 funcrcl 16346 natrcl 16433 ismhm 17160 isghm 17483 isga 17547 isslw 17846 isrhm 18544 rimrcl 18547 islmhm 18848 iscn2 20852 elflim2 21578 isfcls 21623 isnmhm 22360 limcrcl 23444 clwlkswlks 26286 clwwlkprop 26298 iscvm 30495 mclsrcl 30712 ewlkprop 40805 wwlknbp 41044 wspthnp 41048 wwlks2onv 41158 clwwlknbp0 41192 mgmhmrcl 41571 intop 41629 rnghmrcl 41679 rngimrcl 41687 |
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