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Theorem fnrnov 5569
 Description: The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
fnrnov (𝐹 Fn (A × B) → ran 𝐹 = {zx A y B z = (x𝐹y)})
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐹,y,z

Proof of Theorem fnrnov
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 fnrnfv 5145 . 2 (𝐹 Fn (A × B) → ran 𝐹 = {zw (A × B)z = (𝐹w)})
2 fveq2 5103 . . . . . 6 (w = ⟨x, y⟩ → (𝐹w) = (𝐹‘⟨x, y⟩))
3 df-ov 5439 . . . . . 6 (x𝐹y) = (𝐹‘⟨x, y⟩)
42, 3syl6eqr 2072 . . . . 5 (w = ⟨x, y⟩ → (𝐹w) = (x𝐹y))
54eqeq2d 2033 . . . 4 (w = ⟨x, y⟩ → (z = (𝐹w) ↔ z = (x𝐹y)))
65rexxp 4407 . . 3 (w (A × B)z = (𝐹w) ↔ x A y B z = (x𝐹y))
76abbii 2135 . 2 {zw (A × B)z = (𝐹w)} = {zx A y B z = (x𝐹y)}
81, 7syl6eq 2070 1 (𝐹 Fn (A × B) → ran 𝐹 = {zx A y B z = (x𝐹y)})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228  {cab 2008  ∃wrex 2285  ⟨cop 3353   × cxp 4270  ran crn 4273   Fn wfn 4824  ‘cfv 4829  (class class class)co 5436 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-ov 5439 This theorem is referenced by:  ovelrn  5572
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