Theorem br1steqg 7943

Description: Uniqueness condition for the binary relation 1^st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)

Assertion

Ref	Expression
br1steqg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))

Proof of Theorem br1steqg

Step	Hyp	Ref	Expression
1		op1stg 7933	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
2	1	eqeq1d 2733	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ 𝐴 = 𝐶))
3		fo1st 7941	. . . 4 ⊢ 1st :V–onto→V
4		fofn 6737	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
5	3, 4	ax-mp 5	. . 3 ⊢ 1st Fn V
6		opex 5402	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		fnbrfvb 6872	. . 3 ⊢ ((1st Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
8	5, 6, 7	mp2an 692	. 2 ⊢ ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶)
9		eqcom 2738	. 2 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
10	2, 8, 9	3bitr3g 313	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579   class class class wbr 5089   Fn wfn 6476  –onto→wfo 6479  ‘cfv 6481  1^st c1st 7919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921

This theorem is referenced by:  br1steq  35815  fv1stcnv  35821  brxrn  38406