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Theorem foov 5586
 Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
foov (𝐹:(A × B)–onto𝐶 ↔ (𝐹:(A × B)⟶𝐶 z 𝐶 x A y B z = (x𝐹y)))
Distinct variable groups:   x,y,z,A   x,B,y,z   z,𝐶   x,𝐹,y,z
Allowed substitution hints:   𝐶(x,y)

Proof of Theorem foov
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 dffo3 5255 . 2 (𝐹:(A × B)–onto𝐶 ↔ (𝐹:(A × B)⟶𝐶 z 𝐶 w (A × B)z = (𝐹w)))
2 fveq2 5119 . . . . . . 7 (w = ⟨x, y⟩ → (𝐹w) = (𝐹‘⟨x, y⟩))
3 df-ov 5455 . . . . . . 7 (x𝐹y) = (𝐹‘⟨x, y⟩)
42, 3syl6eqr 2087 . . . . . 6 (w = ⟨x, y⟩ → (𝐹w) = (x𝐹y))
54eqeq2d 2048 . . . . 5 (w = ⟨x, y⟩ → (z = (𝐹w) ↔ z = (x𝐹y)))
65rexxp 4422 . . . 4 (w (A × B)z = (𝐹w) ↔ x A y B z = (x𝐹y))
76ralbii 2324 . . 3 (z 𝐶 w (A × B)z = (𝐹w) ↔ z 𝐶 x A y B z = (x𝐹y))
87anbi2i 430 . 2 ((𝐹:(A × B)⟶𝐶 z 𝐶 w (A × B)z = (𝐹w)) ↔ (𝐹:(A × B)⟶𝐶 z 𝐶 x A y B z = (x𝐹y)))
91, 8bitri 173 1 (𝐹:(A × B)–onto𝐶 ↔ (𝐹:(A × B)⟶𝐶 z 𝐶 x A y B z = (x𝐹y)))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300  ∃wrex 2301  ⟨cop 3369   × cxp 4285  ⟶wf 4840  –onto→wfo 4842  ‘cfv 4844  (class class class)co 5452 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-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934 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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-fo 4850  df-fv 4852  df-ov 5455 This theorem is referenced by: (None)
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