MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconst2g Structured version   Visualization version   GIF version

Theorem fconst2g 7188
Description: A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst2g (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Proof of Theorem fconst2g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvconst 7146 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
21adantlr 713 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
3 fvconst2g 7187 . . . . . . 7 ((𝐵𝐶𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
43adantll 712 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
52, 4eqtr4d 2774 . . . . 5 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
65ralrimiva 3145 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
7 ffn 6704 . . . . 5 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8 fnconstg 6766 . . . . 5 (𝐵𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9 eqfnfv 7018 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
107, 8, 9syl2an 596 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
116, 10mpbird 256 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → 𝐹 = (𝐴 × {𝐵}))
1211expcom 414 . 2 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13 fconstg 6765 . . 3 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14 feq1 6685 . . 3 (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
1513, 14syl5ibrcom 246 . 2 (𝐵𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
1612, 15impbid 211 1 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  {csn 4622   × cxp 5667   Fn wfn 6527  wf 6528  cfv 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540
This theorem is referenced by:  fconst2  7190  fconst5  7191  snmapen  9021  repsdf2  14710  cnconst  22717  padct  31815  prv1n  34253  fconst7  43742  eufsnlem  47155  mofsn  47158  mofeu  47162
  Copyright terms: Public domain W3C validator