Theorem ov 5596

Description: The value of an operation class abstraction. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 16-May-1995.) (Revised by set.mm contributors, 24-Jul-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 5527	. . . . 5 ⊢ (AFB) = (F ‘⟨A, B⟩)
2		ov.6	. . . . . 6 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)}
3	2	fveq1i 5330	. . . . 5 ⊢ (F ‘⟨A, B⟩) = ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)} ‘⟨A, B⟩)
4	1, 3	eqtri 2373	. . . 4 ⊢ (AFB) = ({⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)} ‘⟨A, B⟩)
5	4	eqeq1i 2360	. . 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 5587	. . . . 5 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)} Fn {⟨x, y⟩ ∣ (x ∈ R ∧ y ∈ S)}
8		eleq1 2413	. . . . . . . 8 ⊢ (x = A → (x ∈ R ↔ A ∈ R))
9	8	anbi1d 685	. . . . . . 7 ⊢ (x = A → ((x ∈ R ∧ y ∈ S) ↔ (A ∈ R ∧ y ∈ S)))
10		eleq1 2413	. . . . . . . 8 ⊢ (y = B → (y ∈ S ↔ B ∈ S))
11	10	anbi2d 684	. . . . . . 7 ⊢ (y = B → ((A ∈ R ∧ y ∈ S) ↔ (A ∈ R ∧ B ∈ S)))
12	9, 11	opelopabg 4706	. . . . . 6 ⊢ ((A ∈ R ∧ B ∈ S) → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ R ∧ y ∈ S)} ↔ (A ∈ R ∧ B ∈ S)))
13	12	ibir 233	. . . . 5 ⊢ ((A ∈ R ∧ B ∈ S) → ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ R ∧ y ∈ S)})
14		fnopfvb 5360	. . . . 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 644	. . . 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 691	. . . . . 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 691	. . . . . 6 ⊢ (y = B → (((A ∈ R ∧ y ∈ S) ∧ ψ) ↔ ((A ∈ R ∧ B ∈ S) ∧ χ)))
21		ov.4	. . . . . . 7 ⊢ (z = C → (χ ↔ θ))
22	21	anbi2d 684	. . . . . 6 ⊢ (z = C → (((A ∈ R ∧ B ∈ S) ∧ χ) ↔ ((A ∈ R ∧ B ∈ S) ∧ θ)))
23	18, 20, 22	eloprabg 5580	. . . . 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 1266	. . . 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 244	. . 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 248	. 2 ⊢ ((A ∈ R ∧ B ∈ S) → ((AFB) = C ↔ ((A ∈ R ∧ B ∈ S) ∧ θ)))
27	26	bianabs 850	1 ⊢ ((A ∈ R ∧ B ∈ S) → ((AFB) = C ↔ θ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860  ⟨cop 4562  {copab 4623   Fn wfn 4777   ‘cfv 4782  (class class class)co 5526  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-ov 5527  df-oprab 5529

This theorem is referenced by: (None)