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Mirrors > Home > MPE Home > Th. List > Mathboxes > df2ndres | Structured version Visualization version GIF version |
Description: Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
Ref | Expression |
---|---|
df2ndres | ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2nd2 7867 | . . . 4 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) | |
2 | 1 | reseq1i 5847 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) |
3 | resoprab 7328 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} | |
4 | resres 5864 | . . . 4 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) | |
5 | incom 4115 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵)) | |
6 | xpss 5567 | . . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
7 | df-ss 3883 | . . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)) | |
8 | 6, 7 | mpbi 233 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵) |
9 | 5, 8 | eqtr3i 2767 | . . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵) |
10 | 9 | reseq2i 5848 | . . . 4 ⊢ (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵)) |
11 | 4, 10 | eqtri 2765 | . . 3 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵)) |
12 | 2, 3, 11 | 3eqtr3ri 2774 | . 2 ⊢ (2nd ↾ (𝐴 × 𝐵)) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} |
13 | df-mpo 7218 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} | |
14 | 12, 13 | eqtr4i 2768 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 × cxp 5549 ↾ cres 5553 {coprab 7214 ∈ cmpo 7215 2nd c2nd 7760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fo 6386 df-fv 6388 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 |
This theorem is referenced by: cnre2csqima 31575 |
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