Theorem funbrafv2 47710

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6875. (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 6502	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5672	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 586	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5076	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 636	. . . . 5 ⊢ (𝑥 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑥) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2751	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
7	5, 6	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8		funeu 6510	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9		tz6.12-1-afv2 47704	. . . . . 6 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
10	8, 9	sylan2 599	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
11	10	anabss7 679	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
12	7, 11	vtoclg 3500	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
13	3, 12	mpcom 38	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
14	13	ex 413	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃!weu 2572  Vcvv 3431   class class class wbr 5072  Rel wrel 5623  Fun wfun 6479  ''''cafv2 47671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fn 6488  df-dfat 47582  df-afv2 47672

This theorem is referenced by:  fnbrafv2b  47711