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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 3168 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {𝑎 ∈ 𝑁 ∣ 𝜑} = {𝑎 ∈ 𝐿 ∣ 𝜓}) |
7 | 3, 6 | mpteq12dv 4663 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
8 | 7 | adantl 481 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
9 | eqidd 2611 | . 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) ∧ 𝑥 = 𝑋) → V = V) | |
10 | elex 3185 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V) | |
11 | 10 | 3ad2ant1 1075 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑋 ∈ V) |
12 | elex 3185 | . . 3 ⊢ (𝑌 ∈ 𝑊 → 𝑌 ∈ V) | |
13 | 12 | 3ad2ant2 1076 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → 𝑌 ∈ V) |
14 | mptexg 6389 | . . 3 ⊢ (𝐾 ∈ 𝑈 → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V) | |
15 | 14 | 3ad2ant3 1077 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) ∈ V) |
16 | nfv 1830 | . 2 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) | |
17 | nfv 1830 | . 2 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) | |
18 | nfcv 2751 | . 2 ⊢ Ⅎ𝑦𝑋 | |
19 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝑌 | |
20 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑥𝐾 | |
21 | ovmpt3rab1.x | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
22 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥𝐿 | |
23 | 21, 22 | nfrab 3100 | . . 3 ⊢ Ⅎ𝑥{𝑎 ∈ 𝐿 ∣ 𝜓} |
24 | 20, 23 | nfmpt 4674 | . 2 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) |
25 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑦𝐾 | |
26 | ovmpt3rab1.y | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
27 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑦𝐿 | |
28 | 26, 27 | nfrab 3100 | . . 3 ⊢ Ⅎ𝑦{𝑎 ∈ 𝐿 ∣ 𝜓} |
29 | 25, 28 | nfmpt 4674 | . 2 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓}) |
30 | 2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29 | ovmpt2dxf 6684 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐾 ∈ 𝑈) → (𝑋𝑂𝑌) = (𝑧 ∈ 𝐾 ↦ {𝑎 ∈ 𝐿 ∣ 𝜓})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 (class class class)co 6549 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: ovmpt3rabdm 6790 elovmpt3rab1 6791 |
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