Theorem br2ndeqg 7233

Description: Uniqueness condition for the binary relation 2^nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)

Assertion

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

Proof of Theorem br2ndeqg

Step	Hyp	Ref	Expression
1		op2ndg 7223	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
2	1	eqeq1d 2653	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ 𝐵 = 𝐶))
3		fo2nd 7231	. . . 4 ⊢ 2nd :V–onto→V
4		fofn 6155	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
5	3, 4	ax-mp 5	. . 3 ⊢ 2nd Fn V
6		opex 4962	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		fnbrfvb 6274	. . 3 ⊢ ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
8	5, 6, 7	mp2an 708	. 2 ⊢ ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9		eqcom 2658	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
10	2, 8, 9	3bitr3g 302	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685   Fn wfn 5921  –onto→wfo 5924  ‘cfv 5926  2^nd c2nd 7209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-2nd 7211

This theorem is referenced by:  br2ndeq  31797  fv2ndcnv  31805  brxrn  34276