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Mathbox for Thierry Arnoux |
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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 8085 | . . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V)) | |
2 | 1 | reseq1i 5978 | . . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) |
3 | resoprab 7526 | . . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} | |
4 | resres 5995 | . . . 4 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) | |
5 | incom 4202 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵)) | |
6 | xpss 5693 | . . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
7 | df-ss 3966 | . . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)) | |
8 | 6, 7 | mpbi 229 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵) |
9 | 5, 8 | eqtr3i 2763 | . . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵) |
10 | 9 | reseq2i 5979 | . . . 4 ⊢ (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵)) |
11 | 4, 10 | eqtri 2761 | . . 3 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵)) |
12 | 2, 3, 11 | 3eqtr3ri 2770 | . 2 ⊢ (2nd ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} |
13 | df-mpo 7414 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)} | |
14 | 12, 13 | eqtr4i 2764 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 × cxp 5675 ↾ cres 5679 {coprab 7410 ∈ cmpo 7411 2nd c2nd 7974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 |
This theorem is referenced by: cnre2csqima 32891 |
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