Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 7286. (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 7761 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) |
3 | opelxpi 5585 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) | |
4 | eleq2 2898 | . . . . . . 7 ⊢ (dom 𝐹 = (𝐴 × 𝐵) → (〈𝑥, 𝑦〉 ∈ dom 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵))) | |
5 | 3, 4 | syl5ibr 247 | . . . . . 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 408 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐶 ∈ 𝑉) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
8 | 7 | 3impa 1102 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 〈𝑥, 𝑦〉 ∈ dom 𝐹) |
9 | 1 | mpofun 7265 | . . . 4 ⊢ Fun 𝐹 |
10 | funres 6390 | . . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {〈𝑥, 𝑦〉})) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}) |
12 | df-dfat 43195 | . . . 4 ⊢ (𝐹 defAt 〈𝑥, 𝑦〉 ↔ (〈𝑥, 𝑦〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝑥, 𝑦〉}))) | |
13 | aovfundmoveq 43257 | . . . 4 ⊢ (𝐹 defAt 〈𝑥, 𝑦〉 → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) | |
14 | 12, 13 | sylbir 236 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝑥, 𝑦〉})) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
15 | 8, 11, 14 | sylancl 586 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = (𝑥𝐹𝑦)) |
16 | 1 | ovmpt4g 7286 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
17 | 15, 16 | eqtrd 2853 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → ((𝑥𝐹𝑦)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 × cxp 5546 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 (class class class)co 7145 ∈ cmpo 7147 defAt wdfat 43192 ((caov 43194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-aiota 43162 df-dfat 43195 df-afv 43196 df-aov 43197 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |