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Mathbox for Peter Mazsa |
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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 6050 | . 2 ⊢ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
2 | inxprnres 37990 | . 2 ⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | |
3 | 1, 2 | eqtr4i 2757 | 1 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 class class class wbr 5153 {copab 5215 × cxp 5680 ran crn 5683 ↾ cres 5684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 |
This theorem is referenced by: xrnres4 38103 xrnresex 38104 |
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