Theorem df2ndres 33369

Description: Definition for a restriction of the 2^nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)

Assertion

Ref	Expression
df2ndres	⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem df2ndres

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df2nd2 8137	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
2	1	reseq1i 5977	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵))
3		resoprab 7561	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)}
4		resres 5994	. . . 4 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5		incom 4109	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6		xpss 5671	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
7		dfss2 4097	. . . . . . 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 2788	. . . . 5 ⊢ ((V × V) ∩ (𝐴 × 𝐵)) = (𝐴 × 𝐵)
10	9	reseq2i 5978	. . . 4 ⊢ (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ (𝐴 × 𝐵))
11	4, 10	eqtri 2786	. . 3 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (𝐴 × 𝐵))
12	2, 3, 11	3eqtr3ri 2795	. 2 ⊢ (2nd ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)}
13		df-mpo 7448	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)}
14	12, 13	eqtr4i 2789	1 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   × cxp 5653   ↾ cres 5657  {coprab 7444   ∈ cmpo 7445  2^nd c2nd 8024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026

This theorem is used by:  cnre2csqima  34613