Theorem funressnbrafv2 47159

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6971. (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 775	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵 ∈ 𝑊)
2		eleq1 2832	. . . . . . . 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 5170	. . . . . 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 2752	. . . . 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 46962	. . . . . 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 47156	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
13	9, 11, 12	syl2an2 685	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
14	8, 13	vtoclg 3566	. . 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 412	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  {csn 4648   class class class wbr 5166   ↾ cres 5702  Fun wfun 6567  ''''cafv2 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-dfat 47034  df-afv2 47124

This theorem is referenced by:  dfatbrafv2b  47160