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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 7360 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝑅𝑒) = (𝑉𝑅𝐸)) | |
4 | 3 | breqd 5114 | . . . . 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 | sprmpod.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) | |
7 | 6 | 3expb 1120 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → (𝜒 ↔ 𝜓)) |
8 | 5, 7 | anbi12d 631 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 𝑉 ∧ 𝑒 = 𝐸)) → ((𝑥(𝑣𝑅𝑒)𝑦 ∧ 𝜒) ↔ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓))) |
9 | 8 | opabbidv 5169 | . 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 495 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
12 | 10 | simprd 496 | . 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 7402 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝑉𝑅𝐸)𝑦 → 𝜃) ∧ {〈𝑥, 𝑦〉 ∣ 𝜃} ∈ V) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) | |
16 | 13, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)} ∈ V) |
17 | 2, 9, 11, 12, 16 | ovmpod 7501 | 1 ⊢ (𝜑 → (𝑉𝑀𝐸) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝑉𝑅𝐸)𝑦 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 {copab 5165 (class class class)co 7351 ∈ cmpo 7353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 |
This theorem is referenced by: (None) |
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