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Mirrors > Home > MPE Home > Th. List > mpoxeldm | Structured version Visualization version GIF version |
Description: If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.) |
Ref | Expression |
---|---|
mpoxeldm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
Ref | Expression |
---|---|
mpoxeldm | ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxeldm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
2 | 1 | dmmpossx 7814 | . . 3 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷) |
3 | elfvdm 6727 | . . . 4 ⊢ (𝑁 ∈ (𝐹‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐹) | |
4 | df-ov 7194 | . . . 4 ⊢ (𝑋𝐹𝑌) = (𝐹‘〈𝑋, 𝑌〉) | |
5 | 3, 4 | eleq2s 2849 | . . 3 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → 〈𝑋, 𝑌〉 ∈ dom 𝐹) |
6 | 2, 5 | sseldi 3885 | . 2 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → 〈𝑋, 𝑌〉 ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷)) |
7 | nfcsb1v 3823 | . . 3 ⊢ Ⅎ𝑥⦋𝑋 / 𝑥⦌𝐷 | |
8 | csbeq1a 3812 | . . 3 ⊢ (𝑥 = 𝑋 → 𝐷 = ⦋𝑋 / 𝑥⦌𝐷) | |
9 | 7, 8 | opeliunxp2f 7930 | . 2 ⊢ (〈𝑋, 𝑌〉 ∈ ∪ 𝑥 ∈ 𝐶 ({𝑥} × 𝐷) ↔ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)) |
10 | 6, 9 | sylib 221 | 1 ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⦋csb 3798 {csn 4527 〈cop 4533 ∪ ciun 4890 × cxp 5534 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 |
This theorem is referenced by: mpoxneldm 7932 nbgrcl 27377 |
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