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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovmpt4g | Structured version Visualization version GIF version |
Description: Value of a function given by the maps-to notation, analogous to ovmpt4g 7593. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovmpt4g.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
aovmpt4g | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aovmpt4g.3 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | dmmpog 8111 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) |
3 | opelxpi 5736 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) | |
4 | eleq2 2827 | . . . . . . 7 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ dom 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
5 | 3, 4 | imbitrrid 246 | . . . . . 6 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ dom 𝐹)) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ dom 𝐹)) |
7 | 6 | impcom 407 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐶 ∈ 𝑉) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
8 | 7 | 3impa 1110 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
9 | 1 | mpofun 7570 | . . . 4 ⊢ Fun 𝐹 |
10 | funres 6619 | . . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {〈𝑥, 𝑦〉})) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}) |
12 | df-dfat 46967 | . . . 4 ⊢ (𝐹 defAt 〈𝑥, 𝑦〉 ↔ (〈𝑥, 𝑦〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}))) | |
13 | aovfundmoveq 47029 | . . . 4 ⊢ (𝐹 defAt 〈𝑥, 𝑦〉 → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) | |
14 | 12, 13 | sylbir 235 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝑥, 𝑦〉})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
15 | 8, 11, 14 | sylancl 585 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
16 | 1 | ovmpt4g 7593 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
17 | 15, 16 | eqtrd 2774 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 {csn 4648 〈cop 4654 × cxp 5697 dom cdm 5699 ↾ cres 5701 Fun wfun 6566 (class class class)co 7445 ∈ cmpo 7447 defAt wdfat 46964 ((caov 46966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-1st 8026 df-2nd 8027 df-aiota 46933 df-dfat 46967 df-afv 46968 df-aov 46969 |
This theorem is referenced by: (None) |
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