Theorem funressnbrafv2 47490

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6882. (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 2824	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
3	2	anbi2d 630	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)))
4	3	anbi1d 631	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5		breq2 5102	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
6	4, 5	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7		eqeq2 2748	. . . . 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 47293	. . . . . 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 47487	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
13	9, 11, 12	syl2an2 686	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
14	8, 13	vtoclg 3511	. . 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 1541   ∈ wcel 2113  ∃!weu 2568  {csn 4580   class class class wbr 5098   ↾ cres 5626  Fun wfun 6486  ''''cafv2 47454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-dfat 47365  df-afv2 47455

This theorem is referenced by:  dfatbrafv2b  47491