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Mirrors > Home > MPE Home > Th. List > sprmpod | 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 |
---|---|
sprmpod.1 | ⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)}) |
sprmpod.2 | ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) |
sprmpod.3 | ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) |
sprmpod.4 | ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) |
sprmpod.5 | ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) |
Ref | Expression |
---|---|
sprmpod | ⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprmpod.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 7413 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸)) | |
4 | 3 | breqd 5152 | . . . . 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 | sprmpod.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) | |
7 | 6 | 3expb 1117 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓)) |
8 | 5, 7 | anbi12d 630 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓))) |
9 | 8 | opabbidv 5207 | . 2 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
10 | sprmpod.3 | . . 3 ⊢ (𝜑 → (𝑉 ∈ V ∧ 𝐸 ∈ V)) | |
11 | 10 | simpld 494 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
12 | 10 | simprd 495 | . 2 ⊢ (𝜑 → 𝐸 ∈ V) |
13 | sprmpod.4 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃)) | |
14 | sprmpod.5 | . . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) | |
15 | opabbrex 7455 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜃} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) | |
16 | 13, 14, 15 | syl2anc 583 | . 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) |
17 | 2, 9, 11, 12, 16 | ovmpod 7555 | 1 ⊢ (𝜑 → (𝑉𝑀𝐸) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∀wal 1531 = wceq 1533 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 {copab 5203 (class class class)co 7404 ∈ cmpo 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 |
This theorem is referenced by: (None) |
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