Theorem ecoptocl 6652

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 6617	. . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2		eqid 2189	. . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶)
3		eceq1 6598	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
4	3	eqeq2d 2201	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 ↔ 𝐴 = [𝑧]𝑅))
5	4	imbi1d 231	. . . . 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 2192	. . . . . 6 ⊢ (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑 ↔ 𝜓))
9	6, 8	syl5ibcom 155	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → 𝜓))
10	2, 5, 9	optocl 4723	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓))
11	10	rexlimiv 2601	. . 3 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅 → 𝜓)
12	1, 11	syl 14	. 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓)
13		ecoptocl.1	. 2 ⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅)
14	12, 13	eleq2s 2284	1 ⊢ (𝐴 ∈ 𝑆 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∃wrex 2469  ⟨cop 3613   × cxp 4645  [cec 6561   / cqs 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-cnv 4655  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-ec 6565  df-qs 6569

This theorem is referenced by:  2ecoptocl  6653  3ecoptocl  6654  mulidnq  7423  recexnq  7424  ltsonq  7432  distrnq0  7493  addassnq0  7496  ltposr  7797  0idsr  7801  1idsr  7802  00sr  7803  recexgt0sr  7807  archsr  7816  srpospr  7817  map2psrprg  7839