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Theorem eqfnov 5589
 Description: Equality of two operations is determined by their values. (Contributed by set.mm contributors, 1-Sep-2005.)
Assertion
Ref Expression
eqfnov ((F Fn (A × B) G Fn (C × D)) → (F = G ↔ ((A × B) = (C × D) x A y B (xFy) = (xGy))))
Distinct variable groups:   x,y,A   x,B,y   x,F,y   x,G,y
Allowed substitution hints:   C(x,y)   D(x,y)

Proof of Theorem eqfnov
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqfnfv2 5393 . 2 ((F Fn (A × B) G Fn (C × D)) → (F = G ↔ ((A × B) = (C × D) z (A × B)(Fz) = (Gz))))
2 fveq2 5328 . . . . . 6 (z = x, y → (Fz) = (Fx, y))
3 fveq2 5328 . . . . . 6 (z = x, y → (Gz) = (Gx, y))
42, 3eqeq12d 2367 . . . . 5 (z = x, y → ((Fz) = (Gz) ↔ (Fx, y) = (Gx, y)))
5 df-ov 5526 . . . . . 6 (xFy) = (Fx, y)
6 df-ov 5526 . . . . . 6 (xGy) = (Gx, y)
75, 6eqeq12i 2366 . . . . 5 ((xFy) = (xGy) ↔ (Fx, y) = (Gx, y))
84, 7syl6bbr 254 . . . 4 (z = x, y → ((Fz) = (Gz) ↔ (xFy) = (xGy)))
98ralxp 4825 . . 3 (z (A × B)(Fz) = (Gz) ↔ x A y B (xFy) = (xGy))
109anbi2i 675 . 2 (((A × B) = (C × D) z (A × B)(Fz) = (Gz)) ↔ ((A × B) = (C × D) x A y B (xFy) = (xGy)))
111, 10syl6bb 252 1 ((F Fn (A × B) G Fn (C × D)) → (F = G ↔ ((A × B) = (C × D) x A y B (xFy) = (xGy))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  ∀wral 2614  ⟨cop 4561   × cxp 4770   Fn wfn 4776   ‘cfv 4781  (class class class)co 5525 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-csb 3137  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-iun 3971  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 This theorem is referenced by:  eqfnov2  5590
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