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Theorem df1stres 28663
Description: Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Assertion
Ref Expression
df1stres (1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem df1stres
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df1st2 7023 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
21reseq1i 5204 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵))
3 resoprab 6529 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
4 resres 5220 . . . 4 ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5 incom 3670 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6 xpss 5042 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
7 df-ss 3458 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
86, 7mpbi 218 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
95, 8eqtr3i 2538 . . . . 5 ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
109reseq2i 5205 . . . 4 (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × 𝐵))
114, 10eqtri 2536 . . 3 ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ (𝐴 × 𝐵))
122, 3, 113eqtr3ri 2545 . 2 (1st ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
13 df-mpt2 6430 . 2 (𝑥𝐴, 𝑦𝐵𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
1412, 13eqtr4i 2539 1 (1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1938  Vcvv 3077  cin 3443  wss 3444   × cxp 4930  cres 4934  {coprab 6426  cmpt2 6427  1st c1st 6930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-fo 5695  df-fv 5697  df-oprab 6429  df-mpt2 6430  df-1st 6932  df-2nd 6933
This theorem is referenced by:  cnre2csqima  29092
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