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Theorem fconst2g 6373
 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 6336 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
21adantlr 747 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
3 fvconst2g 6372 . . . . . . 7 ((𝐵𝐶𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
43adantll 746 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
52, 4eqtr4d 2647 . . . . 5 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
65ralrimiva 2949 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
7 ffn 5958 . . . . 5 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8 fnconstg 6006 . . . . 5 (𝐵𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9 eqfnfv 6219 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
107, 8, 9syl2an 493 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
116, 10mpbird 246 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → 𝐹 = (𝐴 × {𝐵}))
1211expcom 450 . 2 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13 fconstg 6005 . . 3 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14 feq1 5939 . . 3 (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
1513, 14syl5ibrcom 236 . 2 (𝐵𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
1612, 15impbid 201 1 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125   × cxp 5036   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  fconst2  6375  fconst5  6376  repsdf2  13376  cnconst  20898  padct  28885
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