Theorem qusin 17448

Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusin.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusin.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusin.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusin.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
qusin.s	⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉)

Assertion

Ref	Expression
qusin	⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

Proof of Theorem qusin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusin.s	. . . . 5 ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉)
2		ecinxp 8716	. . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
3	1, 2	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
4	3	mpteq2dva 5182	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))))
5	4	oveq1d 7361	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
6		qusin.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
7		qusin.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
8		eqid 2731	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
9		qusin.e	. . 3 ⊢ (𝜑 → ∼ ∈ 𝑊)
10		qusin.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
11	6, 7, 8, 9, 10	qusval 17446	. 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅))
12		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))
13		eqid 2731	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
14		inex1g 5255	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
15	9, 14	syl 17	. . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
16	12, 7, 13, 15, 10	qusval 17446	. 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
17	5, 11, 16	3eqtr4d 2776	1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897   ↦ cmpt 5170   × cxp 5612   “ cima 5617  ‘cfv 6481  (class class class)co 7346  [cec 8620  Basecbs 17120   “_s cimas 17408   /_s cqus 17409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-ec 8624  df-qus 17413

This theorem is referenced by:  pi1addf  24974  pi1addval  24975  pi1grplem  24976