Theorem fvopab3 2868

Description: Value of a function given by ordered pair abstraction.

Hypotheses

Ref	Expression
fvopab3.1	⊢ B ∈ V
fvopab3.2	⊢ (x = A → (φ ↔ ψ))
fvopab3.3	⊢ (y = B → (ψ ↔ χ))
fvopab3.4	⊢ (x ∈ C → ∃!yφ)
fvopab3.5	⊢ F = {⟨x, y⟩∣(x ∈ C ∧ φ)}

Assertion

Ref	Expression
fvopab3	⊢ (A ∈ C → ((F ‘A) = B ↔ χ))

Distinct variable group(s):   x,y,A   x,B,y   x,C,y   χ,x,y

Proof of Theorem fvopab3

Step	Hyp	Ref	Expression
1		fvopab3.1	. . 3 ⊢ B ∈ V
2		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
3		fvopab3.2	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
4	2, 3	anbi12d 476	. . . 4 ⊢ (x = A → ((x ∈ C ∧ φ) ↔ (A ∈ C ∧ ψ)))
5		fvopab3.3	. . . . 5 ⊢ (y = B → (ψ ↔ χ))
6	5	anbi2d 468	. . . 4 ⊢ (y = B → ((A ∈ C ∧ ψ) ↔ (A ∈ C ∧ χ)))
7	4, 6	opelopabg 2115	. . 3 ⊢ ((A ∈ C ∧ B ∈ V) → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} ↔ (A ∈ C ∧ χ)))
8	1, 7	mpan2 519	. 2 ⊢ (A ∈ C → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} ↔ (A ∈ C ∧ χ)))
9		fvopab3.4	. . . . 5 ⊢ (x ∈ C → ∃!yφ)
10		fvopab3.5	. . . . 5 ⊢ F = {⟨x, y⟩∣(x ∈ C ∧ φ)}
11	9, 10	fnopab 2746	. . . 4 ⊢ F Fn C
12	1	fnfvop 2856	. . . 4 ⊢ ((F Fn C ∧ A ∈ C) → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))
13	11, 12	mpan 518	. . 3 ⊢ (A ∈ C → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))
14	10	eleq2i 1153	. . 3 ⊢ (⟨A, B⟩ ∈ F ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)})
15	13, 14	syl6bb 414	. 2 ⊢ (A ∈ C → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)}))
16		ibar 487	. 2 ⊢ (A ∈ C → (χ ↔ (A ∈ C ∧ χ)))
17	8, 15, 16	3bitr4d 424	1 ⊢ (A ∈ C → ((F ‘A) = B ↔ χ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  {copab 2055   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  recmulpq 3864

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

metamath.org