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Theorem ovigg 5596
 Description: The value of an operation class abstraction. Compare ovig 5597. The condition (x ∈ R ∧ y ∈ S) is been removed. (Contributed by FL, 24-Mar-2007.)
Hypotheses
Ref Expression
ovigg.1 ((x = A y = B z = C) → (φψ))
ovigg.4 ∃*zφ
ovigg.5 F = {x, y, z φ}
Assertion
Ref Expression
ovigg ((A V B W C X) → (ψ → (AFB) = C))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   F(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem ovigg
StepHypRef Expression
1 ovigg.1 . . . 4 ((x = A y = B z = C) → (φψ))
21eloprabga 5578 . . 3 ((A V B W C X) → (A, B, C {x, y, z φ} ↔ ψ))
3 ovigg.4 . . . . 5 ∃*zφ
43funoprab 5584 . . . 4 Fun {x, y, z φ}
5 funopfv 5357 . . . 4 (Fun {x, y, z φ} → (A, B, C {x, y, z φ} → ({x, y, z φ} ‘A, B) = C))
64, 5ax-mp 8 . . 3 (A, B, C {x, y, z φ} → ({x, y, z φ} ‘A, B) = C)
72, 6syl6bir 220 . 2 ((A V B W C X) → (ψ → ({x, y, z φ} ‘A, B) = C))
8 df-ov 5526 . . . 4 (AFB) = (FA, B)
9 ovigg.5 . . . . 5 F = {x, y, z φ}
109fveq1i 5329 . . . 4 (FA, B) = ({x, y, z φ} ‘A, B)
118, 10eqtri 2373 . . 3 (AFB) = ({x, y, z φ} ‘A, B)
1211eqeq1i 2360 . 2 ((AFB) = C ↔ ({x, y, z φ} ‘A, B) = C)
137, 12syl6ibr 218 1 ((A V B W C X) → (ψ → (AFB) = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ⟨cop 4561  Fun wfun 4775   ‘cfv 4781  (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:  ovig  5597  ovmpt2x  5712
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