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Theorem eqfnov 5549
Description: Equality of two operations is determined by their values. (Contributed by NM, 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 5209 . 2 ((𝐹 Fn (A × B) 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((A × B) = (𝐶 × 𝐷) z (A × B)(𝐹z) = (𝐺z))))
2 fveq2 5121 . . . . . 6 (z = ⟨x, y⟩ → (𝐹z) = (𝐹‘⟨x, y⟩))
3 fveq2 5121 . . . . . 6 (z = ⟨x, y⟩ → (𝐺z) = (𝐺‘⟨x, y⟩))
42, 3eqeq12d 2051 . . . . 5 (z = ⟨x, y⟩ → ((𝐹z) = (𝐺z) ↔ (𝐹‘⟨x, y⟩) = (𝐺‘⟨x, y⟩)))
5 df-ov 5458 . . . . . 6 (x𝐹y) = (𝐹‘⟨x, y⟩)
6 df-ov 5458 . . . . . 6 (x𝐺y) = (𝐺‘⟨x, y⟩)
75, 6eqeq12i 2050 . . . . 5 ((x𝐹y) = (x𝐺y) ↔ (𝐹‘⟨x, y⟩) = (𝐺‘⟨x, y⟩))
84, 7syl6bbr 187 . . . 4 (z = ⟨x, y⟩ → ((𝐹z) = (𝐺z) ↔ (x𝐹y) = (x𝐺y)))
98ralxp 4422 . . 3 (z (A × B)(𝐹z) = (𝐺z) ↔ x A y B (x𝐹y) = (x𝐺y))
109anbi2i 430 . 2 (((A × B) = (𝐶 × 𝐷) z (A × B)(𝐹z) = (𝐺z)) ↔ ((A × B) = (𝐶 × 𝐷) x A y B (x𝐹y) = (x𝐺y)))
111, 10syl6bb 185 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 set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wral 2300  cop 3370   × cxp 4286   Fn wfn 4840  cfv 4845  (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:  eqfnov2  5550
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