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Mirrors > Home > MPE Home > Th. List > ovmpt4g | Structured version Visualization version GIF version |
Description: Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 6774.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
ovmpt4g.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
ovmpt4g | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 3506 | . . 3 ⊢ (𝐶 ∈ 𝑉 → ∃𝑧 𝑧 = 𝐶) | |
2 | moeq 3698 | . . . . . . 7 ⊢ ∃*𝑧 𝑧 = 𝐶 | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧 𝑧 = 𝐶) |
4 | ovmpt4g.3 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
5 | df-mpo 7155 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
6 | 4, 5 | eqtri 2844 | . . . . . 6 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
7 | 3, 6 | ovidi 7287 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝑧)) |
8 | eqeq2 2833 | . . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝐶)) | |
9 | 7, 8 | mpbidi 243 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶)) |
10 | 9 | exlimdv 1930 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∃𝑧 𝑧 = 𝐶 → (𝑥𝐹𝑦) = 𝐶)) |
11 | 1, 10 | syl5 34 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶 ∈ 𝑉 → (𝑥𝐹𝑦) = 𝐶)) |
12 | 11 | 3impia 1113 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃*wmo 2616 (class class class)co 7150 {coprab 7151 ∈ cmpo 7152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: ovmpos 7292 ov2gf 7293 ovmpodxf 7294 ovmpodf 7300 ofmres 7679 fnmpoovd 7776 mapxpen 8677 pwfseqlem2 10075 pwfseqlem3 10076 fullfunc 17170 fthfunc 17171 prfcl 17447 curf1cl 17472 curfcl 17476 hofcl 17503 gsum2d2lem 19087 gsum2d2 19088 gsumcom2 19089 dprdval 19119 dprd2d2 19160 cnmpt21 22273 cnmpt2t 22275 cnmptcom 22280 cnmpt2k 22290 xkocnv 22416 suppovss 30420 fedgmullem1 31020 fedgmullem2 31021 fedgmul 31022 madjusmdetlem1 31087 madjusmdetlem3 31089 finxpreclem5 34670 sdclem2 35011 smflimlem1 43040 smflimlem2 43041 aovmpt4g 43393 ovmpordxf 44380 |
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