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Mirrors > Home > MPE Home > Th. List > sprmpt2d | Structured version Visualization version GIF version |
Description: The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 20-Jun-2019.) |
Ref | Expression |
---|---|
sprmpt2d.1 | ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)}) |
sprmpt2d.2 | ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) |
sprmpt2d.3 | ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) |
sprmpt2d.4 | ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) |
sprmpt2d.5 | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) |
Ref | Expression |
---|---|
sprmpt2d | ⊢ (𝜑 → (𝑉𝑀𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprmpt2d.1 | . . 3 ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)})) |
3 | oveq12 6558 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸)) | |
4 | 3 | breqd 4594 | . . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦)) |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦)) |
6 | sprmpt2d.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) | |
7 | 6 | 3expb 1258 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓)) |
8 | 5, 7 | anbi12d 743 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓))) |
9 | 8 | opabbidv 4648 | . 2 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)} = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
10 | sprmpt2d.3 | . . 3 ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
11 | 10 | simpld 474 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
12 | 10 | simprd 478 | . 2 ⊢ (𝜑 → 𝐸 ∈ V) |
13 | sprmpt2d.4 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) | |
14 | sprmpt2d.5 | . . 3 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) | |
15 | opabbrex 6593 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) | |
16 | 13, 14, 15 | syl2anc 691 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) |
17 | 2, 9, 11, 12, 16 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝑉𝑀𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 (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-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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: crcts 26150 cycls 26151 |
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