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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 7597. (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 8115 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) |
3 | opelxpi 5737 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) | |
4 | eleq2 2833 | . . . . . . 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 7574 | . . . 4 ⊢ Fun 𝐹 |
10 | funres 6620 | . . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {〈𝑥, 𝑦〉})) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}) |
12 | df-dfat 47034 | . . . 4 ⊢ (𝐹 defAt 〈𝑥, 𝑦〉 ↔ (〈𝑥, 𝑦〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}))) | |
13 | aovfundmoveq 47096 | . . . 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 7597 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
17 | 15, 16 | eqtrd 2780 | 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 2108 {csn 4648 〈cop 4654 × cxp 5698 dom cdm 5700 ↾ cres 5702 Fun wfun 6567 (class class class)co 7448 ∈ cmpo 7450 defAt wdfat 47031 ((caov 47033 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-aiota 47000 df-dfat 47034 df-afv 47035 df-aov 47036 |
This theorem is referenced by: (None) |
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