Theorem ov 5562

Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ov.1	⊢ 𝐶 ∈ V
ov.2	⊢ (x = A → (φ ↔ ψ))
ov.3	⊢ (y = B → (ψ ↔ χ))
ov.4	⊢ (z = 𝐶 → (χ ↔ θ))
ov.5	⊢ ((x ∈ 𝑅 ∧ y ∈ 𝑆) → ∃!zφ)
ov.6	⊢ 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}

Assertion

Ref	Expression
ov	⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → ((A𝐹B) = 𝐶 ↔ θ))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝑅,y,z   x,𝑆,y,z   θ,x,y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)   𝐹(x,y,z)

Proof of Theorem ov

Step	Hyp	Ref	Expression
1		df-ov 5458	. . . . 5 ⊢ (A𝐹B) = (𝐹‘⟨A, B⟩)
2		ov.6	. . . . . 6 ⊢ 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}
3	2	fveq1i 5122	. . . . 5 ⊢ (𝐹‘⟨A, B⟩) = ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩)
4	1, 3	eqtri 2057	. . . 4 ⊢ (A𝐹B) = ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩)
5	4	eqeq1i 2044	. . 3 ⊢ ((A𝐹B) = 𝐶 ↔ ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩) = 𝐶)
6		ov.5	. . . . . 6 ⊢ ((x ∈ 𝑅 ∧ y ∈ 𝑆) → ∃!zφ)
7	6	fnoprab 5546	. . . . 5 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)} Fn {⟨x, y⟩ ∣ (x ∈ 𝑅 ∧ y ∈ 𝑆)}
8		eleq1 2097	. . . . . . . 8 ⊢ (x = A → (x ∈ 𝑅 ↔ A ∈ 𝑅))
9	8	anbi1d 438	. . . . . . 7 ⊢ (x = A → ((x ∈ 𝑅 ∧ y ∈ 𝑆) ↔ (A ∈ 𝑅 ∧ y ∈ 𝑆)))
10		eleq1 2097	. . . . . . . 8 ⊢ (y = B → (y ∈ 𝑆 ↔ B ∈ 𝑆))
11	10	anbi2d 437	. . . . . . 7 ⊢ (y = B → ((A ∈ 𝑅 ∧ y ∈ 𝑆) ↔ (A ∈ 𝑅 ∧ B ∈ 𝑆)))
12	9, 11	opelopabg 3996	. . . . . 6 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ 𝑅 ∧ y ∈ 𝑆)} ↔ (A ∈ 𝑅 ∧ B ∈ 𝑆)))
13	12	ibir 166	. . . . 5 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ 𝑅 ∧ y ∈ 𝑆)})
14		fnopfvb 5158	. . . . 5 ⊢ (({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)} Fn {⟨x, y⟩ ∣ (x ∈ 𝑅 ∧ y ∈ 𝑆)} ∧ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ 𝑅 ∧ y ∈ 𝑆)}) → (({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}))
15	7, 13, 14	sylancr 393	. . . 4 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → (({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩) = 𝐶 ↔ ⟨⟨A, B⟩, 𝐶⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}))
16		ov.1	. . . . 5 ⊢ 𝐶 ∈ V
17		ov.2	. . . . . . 7 ⊢ (x = A → (φ ↔ ψ))
18	9, 17	anbi12d 442	. . . . . 6 ⊢ (x = A → (((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ) ↔ ((A ∈ 𝑅 ∧ y ∈ 𝑆) ∧ ψ)))
19		ov.3	. . . . . . 7 ⊢ (y = B → (ψ ↔ χ))
20	11, 19	anbi12d 442	. . . . . 6 ⊢ (y = B → (((A ∈ 𝑅 ∧ y ∈ 𝑆) ∧ ψ) ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ χ)))
21		ov.4	. . . . . . 7 ⊢ (z = 𝐶 → (χ ↔ θ))
22	21	anbi2d 437	. . . . . 6 ⊢ (z = 𝐶 → (((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ χ) ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ θ)))
23	18, 20, 22	eloprabg 5534	. . . . 5 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆 ∧ 𝐶 ∈ V) → (⟨⟨A, B⟩, 𝐶⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)} ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ θ)))
24	16, 23	mp3an3 1220	. . . 4 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → (⟨⟨A, B⟩, 𝐶⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)} ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ θ)))
25	15, 24	bitrd 177	. . 3 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → (({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ 𝑅 ∧ y ∈ 𝑆) ∧ φ)}‘⟨A, B⟩) = 𝐶 ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ θ)))
26	5, 25	syl5bb 181	. 2 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → ((A𝐹B) = 𝐶 ↔ ((A ∈ 𝑅 ∧ B ∈ 𝑆) ∧ θ)))
27	26	bianabs 543	1 ⊢ ((A ∈ 𝑅 ∧ B ∈ 𝑆) → ((A𝐹B) = 𝐶 ↔ θ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  Vcvv 2551  ⟨cop 3370  {copab 3808   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455  {coprab 5456

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458  df-oprab 5459

This theorem is referenced by: (None)