Theorem funressnbrafv2 47707

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6883. (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 776	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵 ∈ 𝑊)
2		eleq1 2825	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
3	2	anbi2d 631	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)))
4	3	anbi1d 632	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5		breq2 5090	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴𝐹𝑥 ↔ 𝐴𝐹𝐵))
6	4, 5	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7		eqeq2 2749	. . . . 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 47510	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11	10	3expa 1119	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12		tz6.12-1-afv2 47704	. . . . 5 ⊢ ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
13	9, 11, 12	syl2an2 687	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
14	8, 13	vtoclg 3500	. . 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 1542   ∈ wcel 2114  ∃!weu 2569  {csn 4568   class class class wbr 5086   ↾ cres 5627  Fun wfun 6487  ''''cafv2 47671

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6449  df-fun 6495  df-fn 6496  df-dfat 47582  df-afv2 47672

This theorem is referenced by:  dfatbrafv2b  47708