Theorem fconstg 2775

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

Assertion

Ref	Expression
fconstg	⊢ (B ∈ C → (A × {B}):A–→{B})

Proof of Theorem fconstg

Step	Hyp	Ref	Expression
1		sneq 1816	. . . . 5 ⊢ (x = B → {x} = {B})
2		xpeq2 2441	. . . . 5 ⊢ ({x} = {B} → (A × {x}) = (A × {B}))
3	1, 2	syl 12	. . . 4 ⊢ (x = B → (A × {x}) = (A × {B}))
4	3, 1	jca 236	. . 3 ⊢ (x = B → ((A × {x}) = (A × {B}) ∧ {x} = {B}))
5		feq1 2748	. . . 4 ⊢ ((A × {x}) = (A × {B}) → ((A × {x}):A–→{x} ↔ (A × {B}):A–→{x}))
6		feq3 2750	. . . 4 ⊢ ({x} = {B} → ((A × {B}):A–→{x} ↔ (A × {B}):A–→{B}))
7	5, 6	sylan9bb 418	. . 3 ⊢ (((A × {x}) = (A × {B}) ∧ {x} = {B}) → ((A × {x}):A–→{x} ↔ (A × {B}):A–→{B}))
8	4, 7	syl 12	. 2 ⊢ (x = B → ((A × {x}):A–→{x} ↔ (A × {B}):A–→{B}))
9		visset 1350	. . 3 ⊢ x ∈ V
10	9	fconst 2774	. 2 ⊢ (A × {x}):A–→{x}
11	8, 10	vtoclg 1383	1 ⊢ (B ∈ C → (A × {B}):A–→{B})

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {csn 1808   × cxp 2408  –→wf 2418

This theorem is referenced by:  expp1t 4678  exp1t 4679

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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