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Theorem fconst2 6943
 Description: A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
Hypothesis
Ref Expression
fvconst2.1 𝐵 ∈ V
Assertion
Ref Expression
fconst2 (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Proof of Theorem fconst2
StepHypRef Expression
1 fvconst2.1 . 2 𝐵 ∈ V
2 fconst2g 6941 . 2 (𝐵 ∈ V → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
31, 2ax-mp 5 1 (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3473  {csn 4543   × cxp 5529  ⟶wf 6327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339 This theorem is referenced by:  rrxcph  23975  dvcmul  24526  plyeq0  24787  lnon0  28560  hsn0elch  29010  df0op2  29514  nmop0h  29753  xrge0mulc1cn  31192  matunitlindflem1  34929  poimirlem9  34942  poimir  34966  lfl1  36242  lkr0f  36266  lindsrng01  44668
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