Theorem funfv2 6517

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 6516	. 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
2		funrel 6144	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
3		relimasn 5733	. . . 4 ⊢ (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
4	2, 3	syl 17	. . 3 ⊢ (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
5	4	unieqd 4670	. 2 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
6	1, 5	eqtrd 2861	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   = wceq 1656  {cab 2811  {csn 4399  ∪ cuni 4660   class class class wbr 4875   “ cima 5349  Rel wrel 5351  Fun wfun 6121  ‘cfv 6127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135

This theorem is referenced by:  funfv2f  6518