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Theorem fconst2 7201
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 7199 . 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 1533  wcel 2098  Vcvv 3468  {csn 4623   × cxp 5667  wf 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by:  imadrhmcl  20645  rrxcph  25270  dvcmul  25825  plyeq0  26095  lnon0  30555  hsn0elch  31005  df0op2  31509  nmop0h  31748  xrge0mulc1cn  33450  matunitlindflem1  36996  poimirlem9  37009  poimir  37033  lfl1  38452  lkr0f  38476  lindsrng01  47406
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