Theorem fsn2 5434

Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by set.mm contributors, 19-May-2004.)

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 3760	. . . . 5 ⊢ A ∈ {A}
3		ffvelrn 5415	. . . . 5 ⊢ ((F:{A}–→B ∧ A ∈ {A}) → (F ‘A) ∈ B)
4	2, 3	mpan2 652	. . . 4 ⊢ (F:{A}–→B → (F ‘A) ∈ B)
5		ffn 5223	. . . . 5 ⊢ (F:{A}–→B → F Fn {A})
6		dffn3 5229	. . . . . . 7 ⊢ (F Fn {A} ↔ F:{A}–→ran F)
7	6	biimpi 186	. . . . . 6 ⊢ (F Fn {A} → F:{A}–→ran F)
8		imadmrn 5008	. . . . . . . . 9 ⊢ (F “ dom F) = ran F
9		fndm 5182	. . . . . . . . . 10 ⊢ (F Fn {A} → dom F = {A})
10	9	imaeq2d 4942	. . . . . . . . 9 ⊢ (F Fn {A} → (F “ dom F) = (F “ {A}))
11	8, 10	syl5eqr 2399	. . . . . . . 8 ⊢ (F Fn {A} → ran F = (F “ {A}))
12		fnsnfv 5373	. . . . . . . . 9 ⊢ ((F Fn {A} ∧ A ∈ {A}) → {(F ‘A)} = (F “ {A}))
13	2, 12	mpan2 652	. . . . . . . 8 ⊢ (F Fn {A} → {(F ‘A)} = (F “ {A}))
14	11, 13	eqtr4d 2388	. . . . . . 7 ⊢ (F Fn {A} → ran F = {(F ‘A)})
15		feq3 5212	. . . . . . 7 ⊢ (ran F = {(F ‘A)} → (F:{A}–→ran F ↔ F:{A}–→{(F ‘A)}))
16	14, 15	syl 15	. . . . . 6 ⊢ (F Fn {A} → (F:{A}–→ran F ↔ F:{A}–→{(F ‘A)}))
17	7, 16	mpbid 201	. . . . 5 ⊢ (F Fn {A} → F:{A}–→{(F ‘A)})
18	5, 17	syl 15	. . . 4 ⊢ (F:{A}–→B → F:{A}–→{(F ‘A)})
19	4, 18	jca 518	. . 3 ⊢ (F:{A}–→B → ((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}))
20		snssi 3852	. . . 4 ⊢ ((F ‘A) ∈ B → {(F ‘A)} ⊆ B)
21		fss 5230	. . . . 5 ⊢ ((F:{A}–→{(F ‘A)} ∧ {(F ‘A)} ⊆ B) → F:{A}–→B)
22	21	ancoms 439	. . . 4 ⊢ (({(F ‘A)} ⊆ B ∧ F:{A}–→{(F ‘A)}) → F:{A}–→B)
23	20, 22	sylan 457	. . 3 ⊢ (((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}) → F:{A}–→B)
24	19, 23	impbii 180	. 2 ⊢ (F:{A}–→B ↔ ((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}))
25		fvex 5339	. . . 4 ⊢ (F ‘A) ∈ V
26	1, 25	fsn 5432	. . 3 ⊢ (F:{A}–→{(F ‘A)} ↔ F = {⟨A, (F ‘A)⟩})
27	26	anbi2i 675	. 2 ⊢ (((F ‘A) ∈ B ∧ F:{A}–→{(F ‘A)}) ↔ ((F ‘A) ∈ B ∧ F = {⟨A, (F ‘A)⟩}))
28	24, 27	bitri 240	1 ⊢ (F:{A}–→B ↔ ((F ‘A) ∈ B ∧ F = {⟨A, (F ‘A)⟩}))

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟨cop 4561   “ cima 4722  dom cdm 4772  ran crn 4773   Fn wfn 4776  –→wf 4777   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795

This theorem is referenced by:  fnressn  5438  fressnfv  5439  1cnc  6139