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Theorem eqfnov2 5550
 Description: Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
Assertion
Ref Expression
eqfnov2 ((𝐹 Fn (A × B) 𝐺 Fn (A × B)) → (𝐹 = 𝐺x A y B (x𝐹y) = (x𝐺y)))
Distinct variable groups:   x,A,y   x,B,y   x,𝐹,y   x,𝐺,y

Proof of Theorem eqfnov2
StepHypRef Expression
1 eqfnov 5549 . 2 ((𝐹 Fn (A × B) 𝐺 Fn (A × B)) → (𝐹 = 𝐺 ↔ ((A × B) = (A × B) x A y B (x𝐹y) = (x𝐺y))))
2 simpr 103 . . 3 (((A × B) = (A × B) x A y B (x𝐹y) = (x𝐺y)) → x A y B (x𝐹y) = (x𝐺y))
3 eqidd 2038 . . . 4 (x A y B (x𝐹y) = (x𝐺y) → (A × B) = (A × B))
43ancri 307 . . 3 (x A y B (x𝐹y) = (x𝐺y) → ((A × B) = (A × B) x A y B (x𝐹y) = (x𝐺y)))
52, 4impbii 117 . 2 (((A × B) = (A × B) x A y B (x𝐹y) = (x𝐺y)) ↔ x A y B (x𝐹y) = (x𝐺y))
61, 5syl6bb 185 1 ((𝐹 Fn (A × B) 𝐺 Fn (A × B)) → (𝐹 = 𝐺x A y B (x𝐹y) = (x𝐺y)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300   × cxp 4286   Fn wfn 4840  (class class class)co 5455 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458 This theorem is referenced by:  tpossym  5832
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