Theorem funsi 5521

Description: The singleton image of a function is a function. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
funsi	⊢ (Fun F → Fun SI F)

Proof of Theorem funsi

Dummy variables a b x y c d z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsi 4762	. . . . . . 7 ⊢ (x SI Fy ↔ ∃a∃b(x = {a} ∧ y = {b} ∧ aFb))
2		brsi 4762	. . . . . . 7 ⊢ (x SI Fz ↔ ∃c∃d(x = {c} ∧ z = {d} ∧ cFd))
3	1, 2	anbi12i 678	. . . . . 6 ⊢ ((x SI Fy ∧ x SI Fz) ↔ (∃a∃b(x = {a} ∧ y = {b} ∧ aFb) ∧ ∃c∃d(x = {c} ∧ z = {d} ∧ cFd)))
4		ee4anv 1915	. . . . . 6 ⊢ (∃a∃b∃c∃d((x = {a} ∧ y = {b} ∧ aFb) ∧ (x = {c} ∧ z = {d} ∧ cFd)) ↔ (∃a∃b(x = {a} ∧ y = {b} ∧ aFb) ∧ ∃c∃d(x = {c} ∧ z = {d} ∧ cFd)))
5	3, 4	bitr4i 243	. . . . 5 ⊢ ((x SI Fy ∧ x SI Fz) ↔ ∃a∃b∃c∃d((x = {a} ∧ y = {b} ∧ aFb) ∧ (x = {c} ∧ z = {d} ∧ cFd)))
6		fununiq 5518	. . . . . . . . . . 11 ⊢ ((Fun F ∧ aFb ∧ aFd) → b = d)
7	6	3exp 1150	. . . . . . . . . 10 ⊢ (Fun F → (aFb → (aFd → b = d)))
8		breq1 4643	. . . . . . . . . . . . . . . 16 ⊢ (a = c → (aFd ↔ cFd))
9	8	bicomd 192	. . . . . . . . . . . . . . 15 ⊢ (a = c → (cFd ↔ aFd))
10	9	adantr 451	. . . . . . . . . . . . . 14 ⊢ ((a = c ∧ z = {d}) → (cFd ↔ aFd))
11		eqeq2 2362	. . . . . . . . . . . . . . . 16 ⊢ (z = {d} → ({b} = z ↔ {b} = {d}))
12		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ b ∈ V
13	12	sneqb 3877	. . . . . . . . . . . . . . . 16 ⊢ ({b} = {d} ↔ b = d)
14	11, 13	syl6bb 252	. . . . . . . . . . . . . . 15 ⊢ (z = {d} → ({b} = z ↔ b = d))
15	14	adantl 452	. . . . . . . . . . . . . 14 ⊢ ((a = c ∧ z = {d}) → ({b} = z ↔ b = d))
16	10, 15	imbi12d 311	. . . . . . . . . . . . 13 ⊢ ((a = c ∧ z = {d}) → ((cFd → {b} = z) ↔ (aFd → b = d)))
17	16	biimprcd 216	. . . . . . . . . . . 12 ⊢ ((aFd → b = d) → ((a = c ∧ z = {d}) → (cFd → {b} = z)))
18	17	exp3a 425	. . . . . . . . . . 11 ⊢ ((aFd → b = d) → (a = c → (z = {d} → (cFd → {b} = z))))
19	18	3impd 1165	. . . . . . . . . 10 ⊢ ((aFd → b = d) → ((a = c ∧ z = {d} ∧ cFd) → {b} = z))
20	7, 19	syl6 29	. . . . . . . . 9 ⊢ (Fun F → (aFb → ((a = c ∧ z = {d} ∧ cFd) → {b} = z)))
21		eqeq1 2359	. . . . . . . . . . . . . . . . 17 ⊢ (x = {a} → (x = {c} ↔ {a} = {c}))
22		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ a ∈ V
23	22	sneqb 3877	. . . . . . . . . . . . . . . . 17 ⊢ ({a} = {c} ↔ a = c)
24	21, 23	syl6bb 252	. . . . . . . . . . . . . . . 16 ⊢ (x = {a} → (x = {c} ↔ a = c))
25	24	3anbi1d 1256	. . . . . . . . . . . . . . 15 ⊢ (x = {a} → ((x = {c} ∧ z = {d} ∧ cFd) ↔ (a = c ∧ z = {d} ∧ cFd)))
26	25	adantr 451	. . . . . . . . . . . . . 14 ⊢ ((x = {a} ∧ y = {b}) → ((x = {c} ∧ z = {d} ∧ cFd) ↔ (a = c ∧ z = {d} ∧ cFd)))
27		eqeq1 2359	. . . . . . . . . . . . . . 15 ⊢ (y = {b} → (y = z ↔ {b} = z))
28	27	adantl 452	. . . . . . . . . . . . . 14 ⊢ ((x = {a} ∧ y = {b}) → (y = z ↔ {b} = z))
29	26, 28	imbi12d 311	. . . . . . . . . . . . 13 ⊢ ((x = {a} ∧ y = {b}) → (((x = {c} ∧ z = {d} ∧ cFd) → y = z) ↔ ((a = c ∧ z = {d} ∧ cFd) → {b} = z)))
30	29	imbi2d 307	. . . . . . . . . . . 12 ⊢ ((x = {a} ∧ y = {b}) → ((aFb → ((x = {c} ∧ z = {d} ∧ cFd) → y = z)) ↔ (aFb → ((a = c ∧ z = {d} ∧ cFd) → {b} = z))))
31	30	biimprcd 216	. . . . . . . . . . 11 ⊢ ((aFb → ((a = c ∧ z = {d} ∧ cFd) → {b} = z)) → ((x = {a} ∧ y = {b}) → (aFb → ((x = {c} ∧ z = {d} ∧ cFd) → y = z))))
32	31	exp3a 425	. . . . . . . . . 10 ⊢ ((aFb → ((a = c ∧ z = {d} ∧ cFd) → {b} = z)) → (x = {a} → (y = {b} → (aFb → ((x = {c} ∧ z = {d} ∧ cFd) → y = z)))))
33	32	3impd 1165	. . . . . . . . 9 ⊢ ((aFb → ((a = c ∧ z = {d} ∧ cFd) → {b} = z)) → ((x = {a} ∧ y = {b} ∧ aFb) → ((x = {c} ∧ z = {d} ∧ cFd) → y = z)))
34	20, 33	syl 15	. . . . . . . 8 ⊢ (Fun F → ((x = {a} ∧ y = {b} ∧ aFb) → ((x = {c} ∧ z = {d} ∧ cFd) → y = z)))
35	34	imp3a 420	. . . . . . 7 ⊢ (Fun F → (((x = {a} ∧ y = {b} ∧ aFb) ∧ (x = {c} ∧ z = {d} ∧ cFd)) → y = z))
36	35	exlimdvv 1637	. . . . . 6 ⊢ (Fun F → (∃c∃d((x = {a} ∧ y = {b} ∧ aFb) ∧ (x = {c} ∧ z = {d} ∧ cFd)) → y = z))
37	36	exlimdvv 1637	. . . . 5 ⊢ (Fun F → (∃a∃b∃c∃d((x = {a} ∧ y = {b} ∧ aFb) ∧ (x = {c} ∧ z = {d} ∧ cFd)) → y = z))
38	5, 37	syl5bi 208	. . . 4 ⊢ (Fun F → ((x SI Fy ∧ x SI Fz) → y = z))
39	38	alrimiv 1631	. . 3 ⊢ (Fun F → ∀z((x SI Fy ∧ x SI Fz) → y = z))
40	39	alrimivv 1632	. 2 ⊢ (Fun F → ∀x∀y∀z((x SI Fy ∧ x SI Fz) → y = z))
41		dffun2 5120	. 2 ⊢ (Fun SI F ↔ ∀x∀y∀z((x SI Fy ∧ x SI Fz) → y = z))
42	40, 41	sylibr 203	1 ⊢ (Fun F → Fun SI F)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642  {csn 3738   class class class wbr 4640   SI csi 4721  Fun wfun 4776

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-si 4729  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is used by:  enpw1  6063