Theorem br2ndeqg 8028

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 8018	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
2	1	eqeq1d 2762	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ 𝐵 = 𝐶))
3		fo2nd 8026	. . . 4 ⊢ 2nd :V–onto→V
4		fofn 6803	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
5	3, 4	ax-mp 5	. . 3 ⊢ 2nd Fn V
6		opex 5432	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		fnbrfvb 6940	. . 3 ⊢ ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
8	5, 6, 7	mp2an 705	. 2 ⊢ ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
9		eqcom 2767	. 2 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
10	2, 8, 9	3bitr3g 316	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   class class class wbr 5102   Fn wfn 6538  –onto→wfo 6541  ‘cfv 6543  2^nd c2nd 8004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-2nd 8006

This theorem is used by:  br2ndeq  36632  fv2ndcnv  36638  brxrn  39424