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Theorem fvopab3g 5386
 Description: Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 6-Mar-1996.)
Hypotheses
Ref Expression
fvopab3g.1 B V
fvopab3g.2 (x = A → (φψ))
fvopab3g.3 (y = B → (ψχ))
fvopab3g.4 (x C∃!yφ)
fvopab3g.5 F = {x, y (x C φ)}
Assertion
Ref Expression
fvopab3g (A C → ((FA) = Bχ))
Distinct variable groups:   x,y,A   x,B,y   x,C,y   χ,x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   F(x,y)

Proof of Theorem fvopab3g
StepHypRef Expression
1 fvopab3g.1 . . 3 B V
2 eleq1 2413 . . . . 5 (x = A → (x CA C))
3 fvopab3g.2 . . . . 5 (x = A → (φψ))
42, 3anbi12d 691 . . . 4 (x = A → ((x C φ) ↔ (A C ψ)))
5 fvopab3g.3 . . . . 5 (y = B → (ψχ))
65anbi2d 684 . . . 4 (y = B → ((A C ψ) ↔ (A C χ)))
74, 6opelopabg 4705 . . 3 ((A C B V) → (A, B {x, y (x C φ)} ↔ (A C χ)))
81, 7mpan2 652 . 2 (A C → (A, B {x, y (x C φ)} ↔ (A C χ)))
9 fvopab3g.4 . . . . 5 (x C∃!yφ)
10 fvopab3g.5 . . . . 5 F = {x, y (x C φ)}
119, 10fnopab 5207 . . . 4 F Fn C
12 fnopfvb 5359 . . . 4 ((F Fn C A C) → ((FA) = BA, B F))
1311, 12mpan 651 . . 3 (A C → ((FA) = BA, B F))
1410eleq2i 2417 . . 3 (A, B FA, B {x, y (x C φ)})
1513, 14syl6bb 252 . 2 (A C → ((FA) = BA, B {x, y (x C φ)}))
16 ibar 490 . 2 (A C → (χ ↔ (A C χ)))
178, 15, 163bitr4d 276 1 (A C → ((FA) = Bχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  ⟨cop 4561  {copab 4622   Fn wfn 4776   ‘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-fn 4790  df-fv 4795 This theorem is referenced by: (None)
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