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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 6655 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
3 | 1, 2 | eqtri 2644 | . . . . 5 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
4 | 3 | dmeqi 5325 | . . . 4 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | dmoprabss 6742 | . . . 4 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} ⊆ (𝐴 × 𝐵) | |
6 | 4, 5 | eqsstri 3635 | . . 3 ⊢ dom 𝐹 ⊆ (𝐴 × 𝐵) |
7 | elfvdm 6220 | . . . 4 ⊢ (𝑋 ∈ (𝐹‘〈𝑆, 𝑇〉) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) | |
8 | df-ov 6653 | . . . 4 ⊢ (𝑆𝐹𝑇) = (𝐹‘〈𝑆, 𝑇〉) | |
9 | 7, 8 | eleq2s 2719 | . . 3 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ dom 𝐹) |
10 | 6, 9 | sseldi 3601 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵)) |
11 | opelxp 5146 | . 2 ⊢ (〈𝑆, 𝑇〉 ∈ (𝐴 × 𝐵) ↔ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) | |
12 | 10, 11 | sylib 208 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 〈cop 4183 × cxp 5112 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 {coprab 6651 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: elmpt2cl1 6877 elmpt2cl2 6878 elovmpt2 6879 elovmpt2rab 6880 elovmpt2rab1 6881 el2mpt2csbcl 7250 ixxssixx 12189 funcrcl 16523 natrcl 16610 ismhm 17337 isghm 17660 isga 17724 isslw 18023 isrhm 18721 rimrcl 18724 islmhm 19027 iscn2 21042 elflim2 21768 isfcls 21813 isnmhm 22550 limcrcl 23638 ewlkprop 26499 wwlknbp 26733 wspthnp 26737 wwlks2onv 26847 clwwlknbp0 26884 iscvm 31241 mclsrcl 31458 mgmhmrcl 41781 intop 41839 rnghmrcl 41889 rngimrcl 41897 |
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