Theorem fconst2g 5733

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.

Step	Hyp	Ref	Expression
1		fvconst 5706	. . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
2	1	adantlr 477	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
3		fvconst2g 5732	. . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
4	3	adantll 476	. . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
5	2, 4	eqtr4d 2213	. . . . 5 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
6	5	ralrimiva 2550	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))
7		ffn 5367	. . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴)
8		fnconstg 5415	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) Fn 𝐴)
9		eqfnfv 5615	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
10	7, 8, 9	syl2an 289	. . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)))
11	6, 10	mpbird 167	. . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → 𝐹 = (𝐴 × {𝐵}))
12	11	expcom 116	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵})))
13		fconstg 5414	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
14		feq1 5350	. . 3 ⊢ (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵}))
15	13, 14	syl5ibrcom 157	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵}))
16	12, 15	impbid 129	1 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  {csn 3594   × cxp 4626   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218

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 4123  ax-pow 4176  ax-pr 4211

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 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by:  fconst2  5735  cnconst  13819  nninfall  14843