Theorem fsn2 2896

Description: A function that maps a singleton to a class is the singleton of an ordered pair.

Hypothesis

Ref	Expression
fsn2.1	⊢ A ∈ V

Assertion

Ref	Expression
fsn2	⊢ (F:{A}–→B ↔ ((F ‘A) ∈ B ∧ F = {⟨A, (F ‘A)⟩}))

Proof of Theorem fsn2

Step	Hyp	Ref	Expression
1		fsn2.1	. . . . . 6 ⊢ A ∈ V
2	1	snid 1830	. . . . 5 ⊢ A ∈ {A}
3		ffvrn 2890	. . . . 5 ⊢ ((F:{A}–→B ∧ A ∈ {A}) → (F ‘A) ∈ B)
4	2, 3	mpan2 519	. . . 4 ⊢ (F:{A}–→B → (F ‘A) ∈ B)
5		ffn 2752	. . . . 5 ⊢ (F:{A}–→B → F Fn {A})
6		fnfrn 2758	. . . . . . 7 ⊢ (F Fn {A} ↔ F:{A}–→ran F)
7	6	biimp 133	. . . . . 6 ⊢ (F Fn {A} → F:{A}–→ran F)
8		fndm 2723	. . . . . . . . . 10 ⊢ (F Fn {A} → dom F = {A})
9		imaeq2 2603	. . . . . . . . . 10 ⊢ (dom F = {A} → (F “ dom F) = (F “ {A}))
10	8, 9	syl 12	. . . . . . . . 9 ⊢ (F Fn {A} → (F “ dom F) = (F “ {A}))
11		imadmrn 2610	. . . . . . . . 9 ⊢ (F “ dom F) = ran F
12	10, 11	syl5eqr 1138	. . . . . . . 8 ⊢ (F Fn {A} → ran F = (F “ {A}))
13		fnsnfv 2861	. . . . . . . . 9 ⊢ ((F Fn {A} ∧ A ∈ {A}) → {(F ‘A)} = (F “ {A}))
14	2, 13	mpan2 519	. . . . . . . 8 ⊢ (F Fn {A} → {(F ‘A)} = (F “ {A}))
15	12, 14	eqtr4d 1131	. . . . . . 7 ⊢ (F Fn {A} → ran F = {(F ‘A)})
16		feq3 2750	. . . . . . 7 ⊢ (ran F = {(F ‘A)} → (F:{A}–→ran F ↔ F:{A}–→{(F ‘A)}))
17	15, 16	syl 12	. . . . . 6 ⊢ (F Fn {A} → (F:{A}–→ran F ↔ F:{A}–→{(F ‘A)}))
18	7, 17	mpbid 170	. . . . 5 ⊢ (F Fn {A} → F:{A}–→{(F ‘A)})
19	5, 18	syl 12	. . . 4 ⊢ (F:{A}–→B → F:{A}–→{(F ‘A)})
20	4, 19	jca 236	. . 3 ⊢ (F:{A}–→B → ((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}))
21		fss 2759	. . . . 5 ⊢ ((F:{A}–→{(F ‘A)} ∧ {(F ‘A)} ⊆ B) → F:{A}–→B)
22	21	ancoms 334	. . . 4 ⊢ (({(F ‘A)} ⊆ B ∧ F:{A}–→{(F ‘A)}) → F:{A}–→B)
23		snssi 1851	. . . 4 ⊢ ((F ‘A) ∈ B → {(F ‘A)} ⊆ B)
24	22, 23	sylan 343	. . 3 ⊢ (((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}) → F:{A}–→B)
25	20, 24	impbi 139	. 2 ⊢ (F:{A}–→B ↔ ((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}))
26		fvex 2838	. . . 4 ⊢ (F ‘A) ∈ V
27	1, 26	fsn 2895	. . 3 ⊢ (F:{A}–→{(F ‘A)} ↔ F = {⟨A, (F ‘A)⟩})
28	27	anbi2i 367	. 2 ⊢ (((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}) ↔ ((F ‘A) ∈ B ∧ F = {⟨A, (F ‘A)⟩}))
29	25, 28	bitr 151	1 ⊢ (F:{A}–→B ↔ ((F ‘A) ∈ B ∧ F = {⟨A, (F ‘A)⟩}))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  {csn 1808  ⟨cop 1810  dom cdm 2410  ran crn 2411   “ cima 2413   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  fnressn 2897  fressnfv 2898  en1 3331

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-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  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-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438

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