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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 6659 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸)) | |
4 | 3 | breqd 4664 | . . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑥(𝑣𝑅𝑒)𝑦 ↔ 𝑥(𝑉𝑅𝐸)𝑦)) |
5 | 4 | adantl 482 | . . . 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 1266 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓)) |
8 | 5, 7 | anbi12d 747 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓))) |
9 | 8 | opabbidv 4716 | . 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 475 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
12 | 10 | simprd 479 | . 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 6695 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) | |
16 | 13, 14, 15 | syl2anc 693 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) |
17 | 2, 9, 11, 12, 16 | ovmpt2d 6788 | 1 ⊢ (𝜑 → (𝑉𝑀𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 {copab 4712 (class class class)co 6650 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: (None) |
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