Theorem fnbrafv2b 47803

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6912. (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 47684	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
3	2	funfni 6622	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐹 defAt 𝐵)
4		dfatafv2ex 47768	. . . . . 6 ⊢ (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
5	3, 4	syl 17	. . . . 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 5101	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹''''𝐵)))
8	6, 7	bibi12d 347	. . . . . 6 ⊢ (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
9	8	adantl 485	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10		fneu 6626	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11		tz6.12c-afv2 47797	. . . . . 6 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
12	10, 11	syl 17	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
13	5, 9, 12	vtocld 3526	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
14	1, 13	mpbii 235	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹''''𝐵))
15		breq2 5101	. . 3 ⊢ ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
16	14, 15	syl5ibcom 247	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 → 𝐵𝐹𝐶))
17		fnfun 6616	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18		funbrafv2 47802	. . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
19	17, 18	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
20	19	adantr 484	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
21	16, 20	impbid 214	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃!weu 2594  Vcvv 3453   class class class wbr 5097  Fun wfun 6510   Fn wfn 6511   defAt wdfat 47671  ''''cafv2 47763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-res 5655  df-iota 6472  df-fun 6518  df-fn 6519  df-dfat 47674  df-afv2 47764

This theorem is referenced by:  fnopafv2b  47804  funbrafv22b  47805