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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 7507. (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 8012 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) |
3 | opelxpi 5675 | . . . . . . 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 | syl5ibr 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 409 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹) |
8 | 7 | 3impa 1111 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹) |
9 | 1 | mpofun 7485 | . . . 4 ⊢ Fun 𝐹 |
10 | funres 6548 | . . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}) |
12 | df-dfat 45425 | . . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩}))) | |
13 | aovfundmoveq 45487 | . . . 4 ⊢ (𝐹 defAt ⟨𝑥, 𝑦⟩ → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) | |
14 | 12, 13 | sylbir 234 | . . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝑥, 𝑦⟩})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
15 | 8, 11, 14 | sylancl 587 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
16 | 1 | ovmpt4g 7507 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
17 | 15, 16 | eqtrd 2777 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {csn 4591 ⟨cop 4597 × cxp 5636 dom cdm 5638 ↾ cres 5640 Fun wfun 6495 (class class class)co 7362 ∈ cmpo 7364 defAt wdfat 45422 ((caov 45424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-aiota 45391 df-dfat 45425 df-afv 45426 df-aov 45427 |
This theorem is referenced by: (None) |
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