Theorem fnbrafvb 47874

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6933. (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 6640	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2		eleq2 2852	. . . . . . . 8 ⊢ (𝐴 = dom 𝐹 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
3	2	eqcoms 2771	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ dom 𝐹))
4	3	biimpd 232	. . . . . 6 ⊢ (dom 𝐹 = 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹))
5	1, 4	syl 18	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹))
6	5	imp 411	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
7		snssi 4752	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
8	7	adantl 486	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝐵} ⊆ 𝐴)
9		fnssresb 6659	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
10	9	adantr 485	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
11	8, 10	mpbird 260	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12		fnfun 6637	. . . . 5 ⊢ ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
13	11, 12	syl 18	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → Fun (𝐹 ↾ {𝐵}))
14		df-dfat 47839	. . . . 5 ⊢ (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15		afvfundmfveq 47858	. . . . 5 ⊢ (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹‘𝐵))
16	14, 15	sylbir 238	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹‘𝐵))
17	6, 13, 16	syl2anc 595	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹'''𝐵) = (𝐹‘𝐵))
18	17	eqeq1d 2765	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
19		fnbrfvb 6933	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
20	18, 19	bitrd 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906  {csn 4590   class class class wbr 5110  dom cdm 5663   ↾ cres 5665  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538   defAt wdfat 47836  '''cafv 47837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-aiota 47805  df-dfat 47839  df-afv 47840

This theorem is referenced by:  fnopafvb  47875  funbrafvb  47876  dfafn5a  47880