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Theorem fconst2 7155
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 7153 . 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 205   = wceq 1542  wcel 2107  Vcvv 3444  {csn 4587   × cxp 5632  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  rrxcph  24772  dvcmul  25324  plyeq0  25588  lnon0  29782  hsn0elch  30232  df0op2  30736  nmop0h  30975  xrge0mulc1cn  32579  matunitlindflem1  36120  poimirlem9  36133  poimir  36157  lfl1  37578  lkr0f  37602  imadrhmcl  40761  lindsrng01  46635
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