Theorem df2ndres 33028

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 8095	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2nd ↾ (V × V))
2	1	reseq1i 5976	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵))
3		resoprab 7530	. . 3 ⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝑦)}
4		resres 5993	. . . 4 ⊢ ((2nd ↾ (V × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ ((V × V) ∩ (𝐴 × 𝐵)))
5		incom 4163	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∩ (V × V)) = ((V × V) ∩ (𝐴 × 𝐵))
6		xpss 5679	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (V × V)
7		dfss2 3924	. . . . . . 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 5977	. . . 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 7417	. 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)

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906   × cxp 5661   ↾ cres 5665  {coprab 7413   ∈ cmpo 7414  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988

This theorem is referenced by:  cnre2csqima  34279