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Theorem fconst2g 5319
 Description: A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst2g (B 𝐶 → (𝐹:A⟶{B} ↔ 𝐹 = (A × {B})))

Proof of Theorem fconst2g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fvconst 5294 . . . . . . 7 ((𝐹:A⟶{B} x A) → (𝐹x) = B)
21adantlr 446 . . . . . 6 (((𝐹:A⟶{B} B 𝐶) x A) → (𝐹x) = B)
3 fvconst2g 5318 . . . . . . 7 ((B 𝐶 x A) → ((A × {B})‘x) = B)
43adantll 445 . . . . . 6 (((𝐹:A⟶{B} B 𝐶) x A) → ((A × {B})‘x) = B)
52, 4eqtr4d 2072 . . . . 5 (((𝐹:A⟶{B} B 𝐶) x A) → (𝐹x) = ((A × {B})‘x))
65ralrimiva 2386 . . . 4 ((𝐹:A⟶{B} B 𝐶) → x A (𝐹x) = ((A × {B})‘x))
7 ffn 4989 . . . . 5 (𝐹:A⟶{B} → 𝐹 Fn A)
8 fnconstg 5027 . . . . 5 (B 𝐶 → (A × {B}) Fn A)
9 eqfnfv 5208 . . . . 5 ((𝐹 Fn A (A × {B}) Fn A) → (𝐹 = (A × {B}) ↔ x A (𝐹x) = ((A × {B})‘x)))
107, 8, 9syl2an 273 . . . 4 ((𝐹:A⟶{B} B 𝐶) → (𝐹 = (A × {B}) ↔ x A (𝐹x) = ((A × {B})‘x)))
116, 10mpbird 156 . . 3 ((𝐹:A⟶{B} B 𝐶) → 𝐹 = (A × {B}))
1211expcom 109 . 2 (B 𝐶 → (𝐹:A⟶{B} → 𝐹 = (A × {B})))
13 fconstg 5026 . . 3 (B 𝐶 → (A × {B}):A⟶{B})
14 feq1 4973 . . 3 (𝐹 = (A × {B}) → (𝐹:A⟶{B} ↔ (A × {B}):A⟶{B}))
1513, 14syl5ibrcom 146 . 2 (B 𝐶 → (𝐹 = (A × {B}) → 𝐹:A⟶{B}))
1612, 15impbid 120 1 (B 𝐶 → (𝐹:A⟶{B} ↔ 𝐹 = (A × {B})))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {csn 3367   × cxp 4286   Fn wfn 4840  ⟶wf 4841  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853 This theorem is referenced by:  fconst2  5321
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