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Theorem df1stres 29788
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 7429 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))
21reseq1i 5545 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵))
3 resoprab 6919 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
4 resres 5565 . . . 4 ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5 incom 3946 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6 xpss 5280 . . . . . . 7 (𝐴 × 𝐵) ⊆ (V × V)
7 df-ss 3727 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (V × V) ↔ ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵))
86, 7mpbi 220 . . . . . 6 ((𝐴 × 𝐵) ∩ (V × V)) = (𝐴 × 𝐵)
95, 8eqtr3i 2782 . . . . 5 ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
109reseq2i 5546 . . . 4 (1st ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × 𝐵))
114, 10eqtri 2780 . . 3 ((1st ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (1st ↾ (𝐴 × 𝐵))
122, 3, 113eqtr3ri 2789 . 2 (1st ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
13 df-mpt2 6816 . 2 (𝑥𝐴, 𝑦𝐵𝑥) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝑥)}
1412, 13eqtr4i 2783 1 (1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1630  wcel 2137  Vcvv 3338  cin 3712  wss 3713   × cxp 5262  cres 5266  {coprab 6812  cmpt2 6813  1st c1st 7329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fo 6053  df-fv 6055  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332
This theorem is referenced by:  cnre2csqima  30264
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