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Theorem dffn5 5363
 Description: Representation of a function in terms of its values. (Contributed by set.mm contributors, 29-Jan-2004.)
Assertion
Ref Expression
dffn5 (F Fn AF = {x, y (x A y = (Fx))})
Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem dffn5
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnop 5186 . . . . . . 7 ((F Fn A z, w F) → z A)
21ex 423 . . . . . 6 (F Fn A → (z, w Fz A))
32pm4.71rd 616 . . . . 5 (F Fn A → (z, w F ↔ (z A z, w F)))
4 eqcom 2355 . . . . . . 7 (w = (Fz) ↔ (Fz) = w)
5 fnopfvb 5359 . . . . . . 7 ((F Fn A z A) → ((Fz) = wz, w F))
64, 5syl5bb 248 . . . . . 6 ((F Fn A z A) → (w = (Fz) ↔ z, w F))
76pm5.32da 622 . . . . 5 (F Fn A → ((z A w = (Fz)) ↔ (z A z, w F)))
83, 7bitr4d 247 . . . 4 (F Fn A → (z, w F ↔ (z A w = (Fz))))
9 vex 2862 . . . . 5 z V
10 vex 2862 . . . . 5 w V
11 eleq1 2413 . . . . . 6 (x = z → (x Az A))
12 fveq2 5328 . . . . . . 7 (x = z → (Fx) = (Fz))
1312eqeq2d 2364 . . . . . 6 (x = z → (y = (Fx) ↔ y = (Fz)))
1411, 13anbi12d 691 . . . . 5 (x = z → ((x A y = (Fx)) ↔ (z A y = (Fz))))
15 eqeq1 2359 . . . . . 6 (y = w → (y = (Fz) ↔ w = (Fz)))
1615anbi2d 684 . . . . 5 (y = w → ((z A y = (Fz)) ↔ (z A w = (Fz))))
179, 10, 14, 16opelopab 4708 . . . 4 (z, w {x, y (x A y = (Fx))} ↔ (z A w = (Fz)))
188, 17syl6bbr 254 . . 3 (F Fn A → (z, w Fz, w {x, y (x A y = (Fx))}))
1918eqrelrdv 4852 . 2 (F Fn AF = {x, y (x A y = (Fx))})
20 fvex 5339 . . . 4 (Fx) V
21 eqid 2353 . . . 4 {x, y (x A y = (Fx))} = {x, y (x A y = (Fx))}
2220, 21fnopab2 5208 . . 3 {x, y (x A y = (Fx))} Fn A
23 fneq1 5173 . . 3 (F = {x, y (x A y = (Fx))} → (F Fn A ↔ {x, y (x A y = (Fx))} Fn A))
2422, 23mpbiri 224 . 2 (F = {x, y (x A y = (Fx))} → F Fn A)
2519, 24impbii 180 1 (F Fn AF = {x, y (x A y = (Fx))})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622   Fn wfn 4776   ‘cfv 4781 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 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:  fopabfv  5430  fnov  5591  dffn5v  5706
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