Theorem br1steq 36535

Description: Uniqueness condition for the binary relation 1^st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Hypotheses

Ref	Expression
br1steq.1	⊢ 𝐴 ∈ V
br1steq.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
br1steq	⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)

Proof of Theorem br1steq

Step	Hyp	Ref	Expression
1		br1steq.1	. 2 ⊢ 𝐴 ∈ V
2		br1steq.2	. 2 ⊢ 𝐵 ∈ V
3		br1steqg 8023	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))
4	1, 2, 3	mp2an 705	1 ⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   class class class wbr 5103  1^st c1st 7999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001

This theorem is used by:  dfdm5  36537  brtxp  36642  brpprod  36647  elfuns  36677  brimg  36699  brcup  36701  brcap  36702  brrestrict  36713