Theorem br1steqg 7993

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 7983	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
2	1	eqeq1d 2732	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ 𝐴 = 𝐶))
3		fo1st 7991	. . . 4 ⊢ 1st :V–onto→V
4		fofn 6777	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
5	3, 4	ax-mp 5	. . 3 ⊢ 1st Fn V
6		opex 5427	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		fnbrfvb 6914	. . 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 2737	. 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 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   class class class wbr 5110   Fn wfn 6509  –onto→wfo 6512  ‘cfv 6514  1^st c1st 7969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-1st 7971

This theorem is referenced by:  br1steq  35765  fv1stcnv  35771  brxrn  38363