Theorem funbrafv2 47866

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6927. (Contributed by AV, 6-Sep-2022.)

Assertion

Ref	Expression
funbrafv2	⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Proof of Theorem funbrafv2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 6550	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5713	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 591	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5114	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 641	. . . . 5 ⊢ (𝑥 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑥) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2781	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8		funeu 6558	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9		tz6.12-1-afv2 47860	. . . . . 6 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
10	8, 9	sylan2 604	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
11	10	anabss7 685	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
12	7, 11	vtoclg 3531	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
13	3, 12	mpcom 39	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
14	13	ex 417	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  Vcvv 3463   class class class wbr 5110  Rel wrel 5664  Fun wfun 6527  ''''cafv2 47827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6489  df-fun 6535  df-fn 6536  df-dfat 47738  df-afv2 47828

This theorem is referenced by:  fnbrafv2b  47867