Theorem br2ndeqg 7840

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 7830	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
2	1	eqeq1d 2741	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ 𝐵 = 𝐶))
3		fo2nd 7838	. . . 4 ⊢ 2nd :V–onto→V
4		fofn 6686	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
5	3, 4	ax-mp 5	. . 3 ⊢ 2nd Fn V
6		opex 5381	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		fnbrfvb 6816	. . 3 ⊢ ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
8	5, 6, 7	mp2an 688	. 2 ⊢ ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9		eqcom 2746	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
10	2, 8, 9	3bitr3g 312	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⟨cop 4572   class class class wbr 5078   Fn wfn 6425  –onto→wfo 6428  ‘cfv 6430  2^nd c2nd 7816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-2nd 7818

This theorem is referenced by:  br2ndeq  33725  fv2ndcnv  33731  brxrn  36483