Theorem ovig 5597

Description: The value of an operation class abstraction (weak version). (Contributed by set.mm contributors, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovig.1	⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
ovig.2	⊢ ((x ∈ R ∧ y ∈ S) → ∃*zφ)
ovig.3	⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)}

Assertion

Ref	Expression
ovig	⊢ ((A ∈ R ∧ B ∈ S ∧ C ∈ D) → (ψ → (AFB) = C))

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

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

Proof of Theorem ovig

Step	Hyp	Ref	Expression
1		3simpa 952	. 2 ⊢ ((A ∈ R ∧ B ∈ S ∧ C ∈ D) → (A ∈ R ∧ B ∈ S))
2		eleq1 2413	. . . . . 6 ⊢ (x = A → (x ∈ R ↔ A ∈ R))
3		eleq1 2413	. . . . . 6 ⊢ (y = B → (y ∈ S ↔ B ∈ S))
4	2, 3	bi2anan9 843	. . . . 5 ⊢ ((x = A ∧ y = B) → ((x ∈ R ∧ y ∈ S) ↔ (A ∈ R ∧ B ∈ S)))
5	4	3adant3 975	. . . 4 ⊢ ((x = A ∧ y = B ∧ z = C) → ((x ∈ R ∧ y ∈ S) ↔ (A ∈ R ∧ B ∈ S)))
6		ovig.1	. . . 4 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
7	5, 6	anbi12d 691	. . 3 ⊢ ((x = A ∧ y = B ∧ z = C) → (((x ∈ R ∧ y ∈ S) ∧ φ) ↔ ((A ∈ R ∧ B ∈ S) ∧ ψ)))
8		ovig.2	. . . 4 ⊢ ((x ∈ R ∧ y ∈ S) → ∃*zφ)
9		moanimv 2262	. . . 4 ⊢ (∃*z((x ∈ R ∧ y ∈ S) ∧ φ) ↔ ((x ∈ R ∧ y ∈ S) → ∃*zφ))
10	8, 9	mpbir 200	. . 3 ⊢ ∃*z((x ∈ R ∧ y ∈ S) ∧ φ)
11		ovig.3	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ R ∧ y ∈ S) ∧ φ)}
12	7, 10, 11	ovigg 5596	. 2 ⊢ ((A ∈ R ∧ B ∈ S ∧ C ∈ D) → (((A ∈ R ∧ B ∈ S) ∧ ψ) → (AFB) = C))
13	1, 12	mpand 656	1 ⊢ ((A ∈ R ∧ B ∈ S ∧ C ∈ D) → (ψ → (AFB) = C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  (class class class)co 5525  {coprab 5527

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528

This theorem is referenced by:  ov2ag  5598