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Theorem fvopab3ig 5387
 Description: Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)
Hypotheses
Ref Expression
fvopab3ig.1 (x = A → (φψ))
fvopab3ig.2 (y = B → (ψχ))
fvopab3ig.3 (x C∃*yφ)
fvopab3ig.4 F = {x, y (x C φ)}
Assertion
Ref Expression
fvopab3ig ((A C B D) → (χ → (FA) = B))
Distinct variable groups:   x,y,A   x,B,y   x,C,y   χ,x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   D(x,y)   F(x,y)

Proof of Theorem fvopab3ig
StepHypRef Expression
1 funopab 5139 . . . 4 (Fun {x, y (x C φ)} ↔ x∃*y(x C φ))
2 fvopab3ig.3 . . . . 5 (x C∃*yφ)
3 moanimv 2262 . . . . 5 (∃*y(x C φ) ↔ (x C∃*yφ))
42, 3mpbir 200 . . . 4 ∃*y(x C φ)
51, 4mpgbir 1550 . . 3 Fun {x, y (x C φ)}
6 simpl 443 . . . 4 ((A C B D) → A C)
7 eleq1 2413 . . . . . . 7 (x = A → (x CA C))
8 fvopab3ig.1 . . . . . . 7 (x = A → (φψ))
97, 8anbi12d 691 . . . . . 6 (x = A → ((x C φ) ↔ (A C ψ)))
10 fvopab3ig.2 . . . . . . 7 (y = B → (ψχ))
1110anbi2d 684 . . . . . 6 (y = B → ((A C ψ) ↔ (A C χ)))
129, 11opelopabg 4705 . . . . 5 ((A C B D) → (A, B {x, y (x C φ)} ↔ (A C χ)))
1312biimprd 214 . . . 4 ((A C B D) → ((A C χ) → A, B {x, y (x C φ)}))
146, 13mpand 656 . . 3 ((A C B D) → (χA, B {x, y (x C φ)}))
15 funopfv 5357 . . 3 (Fun {x, y (x C φ)} → (A, B {x, y (x C φ)} → ({x, y (x C φ)} ‘A) = B))
165, 14, 15ee02 1377 . 2 ((A C B D) → (χ → ({x, y (x C φ)} ‘A) = B))
17 fvopab3ig.4 . . . 4 F = {x, y (x C φ)}
1817fveq1i 5329 . . 3 (FA) = ({x, y (x C φ)} ‘A)
1918eqeq1i 2360 . 2 ((FA) = B ↔ ({x, y (x C φ)} ‘A) = B)
2016, 19syl6ibr 218 1 ((A C B D) → (χ → (FA) = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ⟨cop 4561  {copab 4622  Fun wfun 4775   ‘cfv 4781 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 This theorem is referenced by:  fvopab4g  5388  ov6g  5600
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