Theorem fconst 2774

Description: A cross product with a singleton is a constant function.

Hypothesis

Ref	Expression
fconst.1	⊢ B ∈ V

Assertion

Ref	Expression
fconst	⊢ (A × {B}):A–→{B}

Proof of Theorem fconst

Step	Hyp	Ref	Expression
1		f0 2772	. . 3 ⊢ ∅:∅–→{B}
2		xpeq1 2440	. . . . . 6 ⊢ (A = ∅ → (A × {B}) = (∅ × {B}))
3		xp0r 2474	. . . . . 6 ⊢ (∅ × {B}) = ∅
4	2, 3	syl6eq 1140	. . . . 5 ⊢ (A = ∅ → (A × {B}) = ∅)
5		feq1 2748	. . . . 5 ⊢ ((A × {B}) = ∅ → ((A × {B}):A–→{B} ↔ ∅:A–→{B}))
6	4, 5	syl 12	. . . 4 ⊢ (A = ∅ → ((A × {B}):A–→{B} ↔ ∅:A–→{B}))
7		feq2 2749	. . . 4 ⊢ (A = ∅ → (∅:A–→{B} ↔ ∅:∅–→{B}))
8	6, 7	bitrd 406	. . 3 ⊢ (A = ∅ → ((A × {B}):A–→{B} ↔ ∅:∅–→{B}))
9	1, 8	mpbiri 169	. 2 ⊢ (A = ∅ → (A × {B}):A–→{B})
10		dmxp 2552	. . . . . 6 ⊢ (¬ A = ∅ → dom ({B} × A) = {B})
11		df-rn 2429	. . . . . . 7 ⊢ ran (A × {B}) = dom ◡(A × {B})
12		cnvxp 2651	. . . . . . . 8 ⊢ ◡(A × {B}) = ({B} × A)
13	12	dmeqi 2532	. . . . . . 7 ⊢ dom ◡(A × {B}) = dom ({B} × A)
14	11, 13	eqtr 1119	. . . . . 6 ⊢ ran (A × {B}) = dom ({B} × A)
15	10, 14	syl5eq 1136	. . . . 5 ⊢ (¬ A = ∅ → ran (A × {B}) = {B})
16		eqimss 1548	. . . . 5 ⊢ (ran (A × {B}) = {B} → ran (A × {B}) ⊆ {B})
17	15, 16	syl 12	. . . 4 ⊢ (¬ A = ∅ → ran (A × {B}) ⊆ {B})
18		relxp 2486	. . . . . . . 8 ⊢ Rel (A × {B})
19		moeq 1431	. . . . . . . . . . 11 ⊢ ∃*y y = B
20	19	moani 1047	. . . . . . . . . 10 ⊢ ∃*y(x ∈ A ∧ y = B)
21		visset 1350	. . . . . . . . . . . . 13 ⊢ y ∈ V
22	21	brxp 2453	. . . . . . . . . . . 12 ⊢ (x(A × {B})y ↔ (x ∈ A ∧ y ∈ {B}))
23		elsn 1820	. . . . . . . . . . . . 13 ⊢ (y ∈ {B} ↔ y = B)
24	23	anbi2i 367	. . . . . . . . . . . 12 ⊢ ((x ∈ A ∧ y ∈ {B}) ↔ (x ∈ A ∧ y = B))
25	22, 24	bitr 151	. . . . . . . . . . 11 ⊢ (x(A × {B})y ↔ (x ∈ A ∧ y = B))
26	25	bimo 1031	. . . . . . . . . 10 ⊢ (∃*y x(A × {B})y ↔ ∃*y(x ∈ A ∧ y = B))
27	20, 26	mpbir 165	. . . . . . . . 9 ⊢ ∃*y x(A × {B})y
28	27	ax-gen 677	. . . . . . . 8 ⊢ ∀x∃*y x(A × {B})y
29	18, 28	pm3.2i 234	. . . . . . 7 ⊢ (Rel (A × {B}) ∧ ∀x∃*y x(A × {B})y)
30		dffunmo 2679	. . . . . . 7 ⊢ (Fun (A × {B}) ↔ (Rel (A × {B}) ∧ ∀x∃*y x(A × {B})y))
31	29, 30	mpbir 165	. . . . . 6 ⊢ Fun (A × {B})
32		fconst.1	. . . . . . . 8 ⊢ B ∈ V
33	32	snnz 1846	. . . . . . 7 ⊢ ¬ {B} = ∅
34		dmxp 2552	. . . . . . 7 ⊢ (¬ {B} = ∅ → dom (A × {B}) = A)
35	33, 34	ax-mp 6	. . . . . 6 ⊢ dom (A × {B}) = A
36	31, 35	pm3.2i 234	. . . . 5 ⊢ (Fun (A × {B}) ∧ dom (A × {B}) = A)
37		df-fn 2433	. . . . 5 ⊢ ((A × {B}) Fn A ↔ (Fun (A × {B}) ∧ dom (A × {B}) = A))
38	36, 37	mpbir 165	. . . 4 ⊢ (A × {B}) Fn A
39	17, 38	jctil 240	. . 3 ⊢ (¬ A = ∅ → ((A × {B}) Fn A ∧ ran (A × {B}) ⊆ {B}))
40		df-f 2434	. . 3 ⊢ ((A × {B}):A–→{B} ↔ ((A × {B}) Fn A ∧ ran (A × {B}) ⊆ {B}))
41	39, 40	sylibr 175	. 2 ⊢ (¬ A = ∅ → (A × {B}):A–→{B})
42	9, 41	pm2.61i 110	1 ⊢ (A × {B}):A–→{B}

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054   × cxp 2408  ^◡ccnv 2409  dom cdm 2410  ran crn 2411  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  fconstg 2775  fconst2 2902  map0 3268  mapdom2lem 3388  mapdom2 3389  fodomb 3615  clim0 4882  ruclem39 4923  hlim0 5140

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-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-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-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-fun 2432  df-fn 2433  df-f 2434

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