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Theorem fnov 5591
 Description: Representation of an operation class abstraction in terms of its values. (Contributed by set.mm contributors, 7-Feb-2004.)
Assertion
Ref Expression
fnov (F Fn (A × B) ↔ F = {x, y, z ((x A y B) z = (xFy))})
Distinct variable groups:   x,y,z,A   x,B,y,z   x,F,y,z

Proof of Theorem fnov
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 dffn5 5363 . 2 (F Fn (A × B) ↔ F = {w, z (w (A × B) z = (Fw))})
2 elxp 4801 . . . . . . 7 (w (A × B) ↔ xy(w = x, y (x A y B)))
32anbi1i 676 . . . . . 6 ((w (A × B) z = (Fw)) ↔ (xy(w = x, y (x A y B)) z = (Fw)))
4 19.41vv 1902 . . . . . 6 (xy((w = x, y (x A y B)) z = (Fw)) ↔ (xy(w = x, y (x A y B)) z = (Fw)))
5 anass 630 . . . . . . . 8 (((w = x, y (x A y B)) z = (Fw)) ↔ (w = x, y ((x A y B) z = (Fw))))
6 fveq2 5328 . . . . . . . . . . . 12 (w = x, y → (Fw) = (Fx, y))
7 df-ov 5526 . . . . . . . . . . . 12 (xFy) = (Fx, y)
86, 7syl6eqr 2403 . . . . . . . . . . 11 (w = x, y → (Fw) = (xFy))
98eqeq2d 2364 . . . . . . . . . 10 (w = x, y → (z = (Fw) ↔ z = (xFy)))
109anbi2d 684 . . . . . . . . 9 (w = x, y → (((x A y B) z = (Fw)) ↔ ((x A y B) z = (xFy))))
1110pm5.32i 618 . . . . . . . 8 ((w = x, y ((x A y B) z = (Fw))) ↔ (w = x, y ((x A y B) z = (xFy))))
125, 11bitri 240 . . . . . . 7 (((w = x, y (x A y B)) z = (Fw)) ↔ (w = x, y ((x A y B) z = (xFy))))
13122exbii 1583 . . . . . 6 (xy((w = x, y (x A y B)) z = (Fw)) ↔ xy(w = x, y ((x A y B) z = (xFy))))
143, 4, 133bitr2i 264 . . . . 5 ((w (A × B) z = (Fw)) ↔ xy(w = x, y ((x A y B) z = (xFy))))
1514opabbii 4626 . . . 4 {w, z (w (A × B) z = (Fw))} = {w, z xy(w = x, y ((x A y B) z = (xFy)))}
16 dfoprab2 5558 . . . 4 {x, y, z ((x A y B) z = (xFy))} = {w, z xy(w = x, y ((x A y B) z = (xFy)))}
1715, 16eqtr4i 2376 . . 3 {w, z (w (A × B) z = (Fw))} = {x, y, z ((x A y B) z = (xFy))}
1817eqeq2i 2363 . 2 (F = {w, z (w (A × B) z = (Fw))} ↔ F = {x, y, z ((x A y B) z = (xFy))})
191, 18bitri 240 1 (F Fn (A × B) ↔ F = {x, y, z ((x A y B) z = (xFy))})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561  {copab 4622   × cxp 4770   Fn wfn 4776   ‘cfv 4781  (class class class)co 5525  {coprab 5527 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-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-ov 5526  df-oprab 5528 This theorem is referenced by:  fov  5592  fnov2  5707
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