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Mirrors > Home > MPE Home > Th. List > ovmpt3rab1 | Structured version Visualization version GIF version |
Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.) |
Ref | Expression |
---|---|
ovmpt3rab1.o | ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) |
ovmpt3rab1.m | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾) |
ovmpt3rab1.n | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿) |
ovmpt3rab1.p | ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) |
ovmpt3rab1.x | ⊢ Ⅎ𝑥𝜓 |
ovmpt3rab1.y | ⊢ Ⅎ𝑦𝜓 |
Ref | Expression |
---|---|
ovmpt3rab1 | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpt3rab1.o | . . 3 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑})) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))) |
3 | ovmpt3rab1.m | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝐾) | |
4 | ovmpt3rab1.n | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝐿) | |
5 | ovmpt3rab1.p | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | rabeqbidv 3427 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑎 ∈ 𝑁 ∣ 𝜑} = {𝑎 ∈ 𝐿 ∣ 𝜓}) |
7 | 3, 6 | mpteq12dv 5201 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
8 | 7 | adantl 483 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
9 | eqidd 2738 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝑥 = 𝑋) → V = V) | |
10 | elex 3466 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
11 | 10 | 3ad2ant1 1134 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑋 ∈ V) |
12 | elex 3466 | . . 3 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V) | |
13 | 12 | 3ad2ant2 1135 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑌 ∈ V) |
14 | mptexg 7176 | . . 3 ⊢ (𝐾 ∈ 𝑈 → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V) | |
15 | 14 | 3ad2ant3 1136 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V) |
16 | nfv 1918 | . 2 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) | |
17 | nfv 1918 | . 2 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) | |
18 | nfcv 2908 | . 2 ⊢ Ⅎ𝑦𝑋 | |
19 | nfcv 2908 | . 2 ⊢ Ⅎ𝑥𝑌 | |
20 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑥𝐾 | |
21 | ovmpt3rab1.x | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
22 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥𝐿 | |
23 | 21, 22 | nfrabw 3443 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐿 ∣ 𝜓} |
24 | 20, 23 | nfmpt 5217 | . 2 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) |
25 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑦𝐾 | |
26 | ovmpt3rab1.y | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
27 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑦𝐿 | |
28 | 26, 27 | nfrabw 3443 | . . 3 ⊢ Ⅎ𝑦{𝑎 ∈ 𝐿 ∣ 𝜓} |
29 | 25, 28 | nfmpt 5217 | . 2 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) |
30 | 2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29 | ovmpodxf 7510 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 {crab 3410 Vcvv 3448 ↦ cmpt 5193 (class class class)co 7362 ∈ cmpo 7364 |
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-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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-reu 3357 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-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-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 |
This theorem is referenced by: ovmpt3rabdm 7617 elovmpt3rab1 7618 |
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