Theorem ecoptocl 8370

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 8333	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2		eqid 2798	. . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶)
3		eceq1 8310	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
4	3	eqeq2d 2809	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 ↔ 𝐴 = [𝑧]𝑅))
5	4	imbi1d 345	. . . . 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 2806	. . . . . 6 ⊢ (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑 ↔ 𝜓))
9	6, 8	syl5ibcom 248	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → 𝜓))
10	2, 5, 9	optocl 5609	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓))
11	10	rexlimiv 3239	. . 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 2908	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ⟨cop 4531   × cxp 5517  [cec 8270   / cqs 8271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8274  df-qs 8278

This theorem is referenced by:  2ecoptocl  8371  3ecoptocl  8372  0idsr  10508  1idsr  10509  00sr  10510  recexsrlem  10514  map2psrpr  10521