Theorem ecoptocl 8744

Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Hypotheses

Ref	Expression
ecoptocl.1	⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅)
ecoptocl.2	⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))
ecoptocl.3	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)

Assertion

Ref	Expression
ecoptocl	⊢ (𝐴 ∈ 𝑆 → 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑅,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem ecoptocl

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 8703	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2		eqid 2736	. . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶)
3		eceq1 8674	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
4	3	eqeq2d 2747	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 ↔ 𝐴 = [𝑧]𝑅))
5	4	imbi1d 341	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ((𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → 𝜓) ↔ (𝐴 = [𝑧]𝑅 → 𝜓)))
6		ecoptocl.3	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)
7		ecoptocl.2	. . . . . . 7 ⊢ ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑 ↔ 𝜓))
8	7	eqcoms 2744	. . . . . 6 ⊢ (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑 ↔ 𝜓))
9	6, 8	syl5ibcom 245	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → 𝜓))
10	2, 5, 9	optocl 5718	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓))
11	10	rexlimiv 3130	. . 3 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅 → 𝜓)
12	1, 11	syl 17	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓)
13		ecoptocl.1	. 2 ⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅)
14	12, 13	eleq2s 2854	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ⟨cop 4586   × cxp 5622  [cec 8633   / cqs 8634

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 2115  ax-9 2123  ax-ext 2708

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-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641

This theorem is referenced by:  2ecoptocl  8745  3ecoptocl  8746  0idsr  11008  1idsr  11009  00sr  11010  recexsrlem  11014  map2psrpr  11021