Theorem funbrafv2 47695

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6888. (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 6515	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5685	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 581	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5089	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 631	. . . . 5 ⊢ (𝑥 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑥) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2748	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
7	5, 6	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐵 → (((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8		funeu 6523	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9		tz6.12-1-afv2 47689	. . . . . 6 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
10	8, 9	sylan2 594	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
11	10	anabss7 674	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
12	7, 11	vtoclg 3499	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
13	3, 12	mpcom 38	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
14	13	ex 412	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!weu 2568  Vcvv 3429   class class class wbr 5085  Rel wrel 5636  Fun wfun 6492  ''''cafv2 47656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6454  df-fun 6500  df-fn 6501  df-dfat 47567  df-afv2 47657

This theorem is referenced by:  fnbrafv2b  47696