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Mirrors > Home > MPE Home > Th. List > Mathboxes > df1stres | Structured version Visualization version GIF version |
Description: Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
Ref | Expression |
---|---|
df1stres | ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df1st2 7263 | . . . 4 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} = (1st ↾ (V × V)) | |
2 | 1 | reseq1i 5392 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) |
3 | resoprab 6756 | . . 3 ⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)} | |
4 | resres 5409 | . . . 4 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) | |
5 | incom 3805 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵)) | |
6 | xpss 5226 | . . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
7 | df-ss 3588 | . . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)) | |
8 | 6, 7 | mpbi 220 | . . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵) |
9 | 5, 8 | eqtr3i 2646 | . . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵) |
10 | 9 | reseq2i 5393 | . . . 4 ⊢ (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × 𝐵)) |
11 | 4, 10 | eqtri 2644 | . . 3 ⊢ ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ (𝐴 × 𝐵)) |
12 | 2, 3, 11 | 3eqtr3ri 2653 | . 2 ⊢ (1st ↾ (𝐴 × 𝐵)) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)} |
13 | df-mpt2 6655 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑥)} | |
14 | 12, 13 | eqtr4i 2647 | 1 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ↾ cres 5116 {coprab 6651 ↦ cmpt2 6652 1st c1st 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: cnre2csqima 29957 |
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