Theorem fnbrafvb 43343

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6711. (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 6448	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		eleq2 2899	. . . . . . . 8 ⊢ (𝐴 = dom 𝐹 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
3	2	eqcoms 2827	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
4	3	biimpd 231	. . . . . 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 409	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
7		snssi 4733	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
8	7	adantl 484	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
9		fnssresb 6462	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
10	9	adantr 483	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
11	8, 10	mpbird 259	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12		fnfun 6446	. . . . 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 43308	. . . . 5 ⊢ (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15		afvfundmfveq 43327	. . . . 5 ⊢ (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹‘𝐵))
16	14, 15	sylbir 237	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹‘𝐵))
17	6, 13, 16	syl2anc 586	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) = (𝐹‘𝐵))
18	17	eqeq1d 2821	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
19		fnbrfvb 6711	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
20	18, 19	bitrd 281	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108   ⊆ wss 3934  {csn 4559   class class class wbr 5057  dom cdm 5548   ↾ cres 5550  Fun wfun 6342   Fn wfn 6343  ‘cfv 6348   defAt wdfat 43305  '''cafv 43306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-aiota 43275  df-dfat 43308  df-afv 43309

This theorem is referenced by:  fnopafvb  43344  funbrafvb  43345  dfafn5a  43349