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Theorem fconst2g 5731
Description: A constant function expressed as a cross 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 5704 . . . . . . 7 ((𝐹:𝐴⟶{𝐵} ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
21adantlr 477 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = 𝐵)
3 fvconst2g 5730 . . . . . . 7 ((𝐵𝐶𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
43adantll 476 . . . . . 6 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
52, 4eqtr4d 2213 . . . . 5 (((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) ∧ 𝑥𝐴) → (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
65ralrimiva 2550 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥))
7 ffn 5365 . . . . 5 (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8 fnconstg 5413 . . . . 5 (𝐵𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9 eqfnfv 5613 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
107, 8, 9syl2an 289 . . . 4 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥𝐴 (𝐹𝑥) = ((𝐴 × {𝐵})‘𝑥)))
116, 10mpbird 167 . . 3 ((𝐹:𝐴⟶{𝐵} ∧ 𝐵𝐶) → 𝐹 = (𝐴 × {𝐵}))
1211expcom 116 . 2 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13 fconstg 5412 . . 3 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14 feq1 5348 . . 3 (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
1513, 14syl5ibrcom 157 . 2 (𝐵𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
1612, 15impbid 129 1 (𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  {csn 3592   × cxp 4624   Fn wfn 5211  wf 5212  cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224
This theorem is referenced by:  fconst2  5733  cnconst  13665  nninfall  14678
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