![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfres4 | Structured version Visualization version GIF version |
Description: Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.) |
Ref | Expression |
---|---|
dfres4 | ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfres2 5694 | . 2 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
2 | inxprnres 34606 | . 2 ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
3 | 1, 2 | eqtr4i 2852 | 1 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∩ cin 3797 class class class wbr 4875 {copab 4937 × cxp 5344 ran crn 5347 ↾ cres 5348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 |
This theorem is referenced by: xrnres4 34706 xrnresex 34707 |
Copyright terms: Public domain | W3C validator |