Theorem funressnbrafv2 45942

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6942. (Contributed by AV, 7-Sep-2022.)

Assertion

Ref	Expression
funressnbrafv2	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Proof of Theorem funressnbrafv2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 774	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵 ∈ 𝑊)
2		eleq1 2821	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
3	2	anbi2d 629	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)))
4	3	anbi1d 630	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5		breq2 5152	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
6	4, 5	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7		eqeq2 2744	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
8	6, 7	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐵 → (((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
9		id 22	. . . . 5 ⊢ (𝐴𝐹𝑥 → 𝐴𝐹𝑥)
10		funressneu 45747	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11	10	3expa 1118	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12		tz6.12-1-afv2 45939	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
13	9, 11, 12	syl2an2 684	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
14	8, 13	vtoclg 3556	. . 3 ⊢ (𝐵 ∈ 𝑊 → ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
15	1, 14	mpcom 38	. 2 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
16	15	ex 413	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃!weu 2562  {csn 4628   class class class wbr 5148   ↾ cres 5678  Fun wfun 6537  ''''cafv2 45906

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 45817  df-afv2 45907

This theorem is referenced by:  dfatbrafv2b  45943