Theorem fnbrafv2b 48287

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6933. (Contributed by AV, 6-Sep-2022.)

Assertion

Ref	Expression
fnbrafv2b	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Proof of Theorem fnbrafv2b

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (𝐹''''𝐵) = (𝐹''''𝐵)
2		fundmdfat 48168	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
3	2	funfni 6643	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐹 defAt 𝐵)
4		dfatafv2ex 48252	. . . . . 6 ⊢ (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
5	3, 4	syl 18	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ V)
6		eqeq2 2773	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7		breq2 5107	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹''''𝐵)))
8	6, 7	bibi12d 348	. . . . . 6 ⊢ (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
9	8	adantl 487	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10		fneu 6647	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11		tz6.12c-afv2 48281	. . . . . 6 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
12	10, 11	syl 18	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
13	5, 9, 12	vtocld 3523	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
14	1, 13	mpbii 236	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹''''𝐵))
15		breq2 5107	. . 3 ⊢ ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
16	14, 15	syl5ibcom 248	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 → 𝐵𝐹𝐶))
17		fnfun 6637	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18		funbrafv2 48286	. . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
19	17, 18	syl 18	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
20	19	adantr 486	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
21	16, 20	impbid 215	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃!weu 2594  Vcvv 3451   class class class wbr 5103  Fun wfun 6531   Fn wfn 6532   defAt wdfat 48155  ''''cafv2 48247

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-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  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-nfc 2910  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-res 5663  df-iota 6493  df-fun 6539  df-fn 6540  df-dfat 48158  df-afv2 48248

This theorem is used by:  fnopafv2b  48288  funbrafv22b  48289