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Mirrors > Home > MPE Home > Th. List > Mathboxes > ecres2 | Structured version Visualization version GIF version |
Description: The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.) |
Ref | Expression |
---|---|
ecres2 | ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elecres 34592 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦))) | |
2 | 1 | elv 3418 | . . . 4 ⊢ (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑦)) |
3 | 2 | baib 531 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝑦 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ 𝐵𝑅𝑦)) |
4 | 3 | abbi2dv 2947 | . 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = {𝑦 ∣ 𝐵𝑅𝑦}) |
5 | dfec2 8017 | . 2 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 = {𝑦 ∣ 𝐵𝑅𝑦}) | |
6 | 4, 5 | eqtr4d 2864 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {cab 2811 Vcvv 3414 class class class wbr 4875 ↾ cres 5348 [cec 8012 |
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-sbc 3663 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 df-ima 5359 df-ec 8016 |
This theorem is referenced by: eccnvepres2 34599 eldmqsres 34601 qsresid 34643 ecex2 34647 |
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