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Mirrors > Home > MPE Home > Th. List > dfid4 | Structured version Visualization version GIF version |
Description: The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
Ref | Expression |
---|---|
dfid4 | ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcom 1945 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 527 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
4 | 1, 3 | bitri 264 | . . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
5 | 4 | opabbii 4717 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} |
6 | df-id 5024 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
7 | df-mpt 4730 | . 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} | |
8 | 5, 6, 7 | 3eqtr4i 2654 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {copab 4712 ↦ cmpt 4729 I cid 5023 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-opab 4713 df-mpt 4730 df-id 5024 |
This theorem is referenced by: dfid5 13767 dfid6 13768 dfid7 37919 |
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