Theorem fnbrafvb 39883

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6146. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
fnbrafvb	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Proof of Theorem fnbrafvb

Step	Hyp	Ref	Expression
1		fndm 5904	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		eleq2 2677	. . . . . . . 8 ⊢ (𝐴 = dom 𝐹 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
3	2	eqcoms 2618	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
4	3	biimpd 218	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹))
5	1, 4	syl 17	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹))
6	5	imp 444	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
7		snssi 4280	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
8	7	adantl 481	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
9		fnssresb 5917	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
10	9	adantr 480	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
11	8, 10	mpbird 246	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12		fnfun 5902	. . . . 5 ⊢ ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
13	11, 12	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → Fun (𝐹 ↾ {𝐵}))
14		df-dfat 39845	. . . . 5 ⊢ (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15		afvfundmfveq 39867	. . . . 5 ⊢ (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹‘𝐵))
16	14, 15	sylbir 224	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹‘𝐵))
17	6, 13, 16	syl2anc 691	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) = (𝐹‘𝐵))
18	17	eqeq1d 2612	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
19		fnbrfvb 6146	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
20	18, 19	bitrd 267	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804   defAt wdfat 39842  '''cafv 39843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-dfat 39845  df-afv 39846

This theorem is referenced by:  fnopafvb  39884  funbrafvb  39885  dfafn5a  39889