Theorem funfv2 6965

Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)

Assertion

Ref	Expression
funfv2	⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funfv2

Step	Hyp	Ref	Expression
1		funfv 6964	. 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
2		funrel 6548	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
3		relimasn 6079	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
4	2, 3	syl 18	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
5	4	unieqd 4880	. 2 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
6	1, 5	eqtrd 2796	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2739  {csn 4584  ∪ cuni 4867   class class class wbr 5103   “ cima 5654  Rel wrel 5656  Fun wfun 6525  ‘cfv 6531

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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-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-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is used by:  funfv2f  6966