Theorem fconst2 2902

Description: A constant function expressed as a cross product. See also fconstfv 2903.

Hypothesis

Ref	Expression
fconst2.1	⊢ B ∈ V

Assertion

Ref	Expression
fconst2	⊢ (F:A–→{B} ↔ F = (A × {B}))

Proof of Theorem fconst2

Step	Hyp	Ref	Expression
1		fvconst 2899	. . . . . . 7 ⊢ ((F:A–→{B} ∧ x ∈ A) → (F ‘x) = B)
2		fconst2.1	. . . . . . . . . 10 ⊢ B ∈ V
3	2	fconst 2774	. . . . . . . . 9 ⊢ (A × {B}):A–→{B}
4		fvconst 2899	. . . . . . . . 9 ⊢ (((A × {B}):A–→{B} ∧ x ∈ A) → ((A × {B}) ‘x) = B)
5	3, 4	mpan 518	. . . . . . . 8 ⊢ (x ∈ A → ((A × {B}) ‘x) = B)
6	5	adantl 305	. . . . . . 7 ⊢ ((F:A–→{B} ∧ x ∈ A) → ((A × {B}) ‘x) = B)
7	1, 6	eqtr4d 1131	. . . . . 6 ⊢ ((F:A–→{B} ∧ x ∈ A) → (F ‘x) = ((A × {B}) ‘x))
8	7	exp 291	. . . . 5 ⊢ (F:A–→{B} → (x ∈ A → (F ‘x) = ((A × {B}) ‘x)))
9	8	r19.21aiv 1259	. . . 4 ⊢ (F:A–→{B} → ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))
10		cleqid 1102	. . . 4 ⊢ A = A
11	9, 10	jctil 240	. . 3 ⊢ (F:A–→{B} → (A = A ∧ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x)))
12		ffn 2752	. . . 4 ⊢ (F:A–→{B} → F Fn A)
13		ffn 2752	. . . . . 6 ⊢ ((A × {B}):A–→{B} → (A × {B}) Fn A)
14	3, 13	ax-mp 6	. . . . 5 ⊢ (A × {B}) Fn A
15		cleqfv 2880	. . . . 5 ⊢ ((F Fn A ∧ (A × {B}) Fn A) → (F = (A × {B}) ↔ (A = A ∧ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))))
16	14, 15	mpan2 519	. . . 4 ⊢ (F Fn A → (F = (A × {B}) ↔ (A = A ∧ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))))
17	12, 16	syl 12	. . 3 ⊢ (F:A–→{B} → (F = (A × {B}) ↔ (A = A ∧ ∀x ∈ A (F ‘x) = ((A × {B}) ‘x))))
18	11, 17	mpbird 171	. 2 ⊢ (F:A–→{B} → F = (A × {B}))
19		feq1 2748	. . 3 ⊢ (F = (A × {B}) → (F:A–→{B} ↔ (A × {B}):A–→{B}))
20	3, 19	mpbiri 169	. 2 ⊢ (F = (A × {B}) → F:A–→{B})
21	18, 20	impbi 139	1 ⊢ (F:A–→{B} ↔ F = (A × {B}))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  {csn 1808   × cxp 2408   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  map1 3335  hsn0elch 5155  ho0 5612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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