Theorem funbrafv2 48424

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6936. (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 6559	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 5704	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
3	1, 2	sylan 592	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → 𝐵 ∈ V)
4		breq2 5106	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
5	4	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝐵 → ((Fun 𝐹 ∧ 𝐴𝐹𝑥) ↔ (Fun 𝐹 ∧ 𝐴𝐹𝐵)))
6		eqeq2 2772	. . . . 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 6568	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9		tz6.12-1-afv2 48418	. . . . . 6 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
10	8, 9	sylan2 605	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ (Fun 𝐹 ∧ 𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
11	10	anabss7 686	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
12	7, 11	vtoclg 3517	. . 3 ⊢ (𝐵 ∈ V → ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
13	3, 12	mpcom 39	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
14	13	ex 418	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  Vcvv 3450   class class class wbr 5102  Rel wrel 5655  Fun wfun 6535  ''''cafv2 48385

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6496  df-fun 6543  df-fn 6544  df-dfat 48296  df-afv2 48386

This theorem is used by:  fnbrafv2b  48425