Theorem fnbrafv2b 47163

Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6973. (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 2740	. . . 4 ⊢ (𝐹''''𝐵) = (𝐹''''𝐵)
2		fundmdfat 47044	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
3	2	funfni 6685	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐹 defAt 𝐵)
4		dfatafv2ex 47128	. . . . . 6 ⊢ (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
5	3, 4	syl 17	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹''''𝐵) ∈ V)
6		eqeq2 2752	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7		breq2 5170	. . . . . . 7 ⊢ (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹''''𝐵)))
8	6, 7	bibi12d 345	. . . . . 6 ⊢ (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
9	8	adantl 481	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10		fneu 6689	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11		tz6.12c-afv2 47157	. . . . . 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 3573	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
14	1, 13	mpbii 233	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹''''𝐵))
15		breq2 5170	. . 3 ⊢ ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
16	14, 15	syl5ibcom 245	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 → 𝐵𝐹𝐶))
17		fnfun 6679	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
18		funbrafv2 47162	. . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
19	17, 18	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
20	19	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
21	16, 20	impbid 212	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹''''𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  Vcvv 3488   class class class wbr 5166  Fun wfun 6567   Fn wfn 6568   defAt wdfat 47031  ''''cafv2 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-dfat 47034  df-afv2 47124

This theorem is referenced by:  fnopafv2b  47164  funbrafv22b  47165