Theorem br2ndeqg 30920

Description: Uniqueness condition for binary relationship over the 2^nd relationship. (Contributed by Scott Fenton, 2-Jul-2020.)

Assertion

Ref	Expression
br2ndeqg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))

Proof of Theorem br2ndeqg

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4340	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	breq1d 4593	. . . . 5 ⊢ (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝑦⟩2nd 𝐶))
3	2	bibi1d 332	. . . 4 ⊢ (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦) ↔ (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦)))
4	3	imbi2d 329	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ 𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦)) ↔ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦))))
5		opeq2 4341	. . . . . 6 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	breq1d 4593	. . . . 5 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
7		eqeq2 2621	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 = 𝑦 ↔ 𝐶 = 𝐵))
8	6, 7	bibi12d 334	. . . 4 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦) ↔ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)))
9	8	imbi2d 329	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ 𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦)) ↔ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))))
10		breq2 4587	. . . 4 ⊢ (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩2nd 𝑧 ↔ ⟨𝑥, 𝑦⟩2nd 𝐶))
11		eqeq1 2614	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = 𝑦 ↔ 𝐶 = 𝑦))
12		vex 3176	. . . . 5 ⊢ 𝑥 ∈ V
13		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
14		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
15	12, 13, 14	br2ndeq 30918	. . . 4 ⊢ (⟨𝑥, 𝑦⟩2nd 𝑧 ↔ 𝑧 = 𝑦)
16	10, 11, 15	vtoclbg 3240	. . 3 ⊢ (𝐶 ∈ 𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶 ↔ 𝐶 = 𝑦))
17	4, 9, 16	vtocl2g 3243	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)))
18	17	3impia 1253	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-2nd 7060

This theorem is referenced by:  fv2ndcnv  30926