Theorem funbrafv2 47217

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6937. (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 6563	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5719	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 580	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5127	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑥) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2746	. . . . 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 6571	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9		tz6.12-1-afv2 47211	. . . . . 6 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
10	8, 9	sylan2 593	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
11	10	anabss7 673	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
12	7, 11	vtoclg 3537	. . 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 1539   ∈ wcel 2107  ∃!weu 2566  Vcvv 3463   class class class wbr 5123  Rel wrel 5670  Fun wfun 6535  ''''cafv2 47178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-dfat 47089  df-afv2 47179

This theorem is referenced by:  fnbrafv2b  47218