Theorem ecoptocl 8040

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 8003	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2		eqid 2765	. . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶)
3		eceq1 7985	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
4	3	eqeq2d 2775	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 ↔ 𝐴 = [𝑧]𝑅))
5	4	imbi1d 332	. . . . 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 2773	. . . . . 6 ⊢ (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑 ↔ 𝜓))
9	6, 8	syl5ibcom 236	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → 𝜓))
10	2, 5, 9	optocl 5365	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓))
11	10	rexlimiv 3174	. . 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 2862	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1652   ∈ wcel 2155  ∃wrex 3056  ⟨cop 4340   × cxp 5275  [cec 7945   / cqs 7946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ec 7949  df-qs 7953

This theorem is referenced by:  2ecoptocl  8041  3ecoptocl  8042  0idsr  10171  1idsr  10172  00sr  10173  recexsrlem  10177  map2psrpr  10184