Theorem fnopfv 7070

Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Assertion

Ref	Expression
fnopfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ 𝐹)

Proof of Theorem fnopfv

Step	Hyp	Ref	Expression
1		funfvop 7044	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ 𝐹)
2	1	funfni 6640	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ⟨cop 4590   Fn wfn 6529  ‘cfv 6534

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-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6490  df-fun 6536  df-fn 6537  df-fv 6542

This theorem is used by:  foeqcnvco  7303