Theorem dff13 5472

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by set.mm contributors, 29-Oct-1996.)

Assertion

Ref	Expression
dff13	⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Distinct variable groups:   x,y,A   x,F,y

Allowed substitution hints:   B(x, y)

Proof of Theorem dff13

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff12 5258	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀z∃*x xFz))
2		ffn 5224	. . . 4 ⊢ (F:A–→B → F Fn A)
3		breldm 4912	. . . . . . . . . . . . . 14 ⊢ (xFz → x ∈ dom F)
4		fndm 5183	. . . . . . . . . . . . . . 15 ⊢ (F Fn A → dom F = A)
5	4	eleq2d 2420	. . . . . . . . . . . . . 14 ⊢ (F Fn A → (x ∈ dom F ↔ x ∈ A))
6	3, 5	syl5ib 210	. . . . . . . . . . . . 13 ⊢ (F Fn A → (xFz → x ∈ A))
7		breldm 4912	. . . . . . . . . . . . . 14 ⊢ (yFz → y ∈ dom F)
8	4	eleq2d 2420	. . . . . . . . . . . . . 14 ⊢ (F Fn A → (y ∈ dom F ↔ y ∈ A))
9	7, 8	syl5ib 210	. . . . . . . . . . . . 13 ⊢ (F Fn A → (yFz → y ∈ A))
10	6, 9	anim12d 546	. . . . . . . . . . . 12 ⊢ (F Fn A → ((xFz ∧ yFz) → (x ∈ A ∧ y ∈ A)))
11	10	pm4.71rd 616	. . . . . . . . . . 11 ⊢ (F Fn A → ((xFz ∧ yFz) ↔ ((x ∈ A ∧ y ∈ A) ∧ (xFz ∧ yFz))))
12		eqcom 2355	. . . . . . . . . . . . . . 15 ⊢ (z = (F ‘x) ↔ (F ‘x) = z)
13		fnbrfvb 5359	. . . . . . . . . . . . . . 15 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = z ↔ xFz))
14	12, 13	syl5bb 248	. . . . . . . . . . . . . 14 ⊢ ((F Fn A ∧ x ∈ A) → (z = (F ‘x) ↔ xFz))
15		eqcom 2355	. . . . . . . . . . . . . . 15 ⊢ (z = (F ‘y) ↔ (F ‘y) = z)
16		fnbrfvb 5359	. . . . . . . . . . . . . . 15 ⊢ ((F Fn A ∧ y ∈ A) → ((F ‘y) = z ↔ yFz))
17	15, 16	syl5bb 248	. . . . . . . . . . . . . 14 ⊢ ((F Fn A ∧ y ∈ A) → (z = (F ‘y) ↔ yFz))
18	14, 17	bi2anan9 843	. . . . . . . . . . . . 13 ⊢ (((F Fn A ∧ x ∈ A) ∧ (F Fn A ∧ y ∈ A)) → ((z = (F ‘x) ∧ z = (F ‘y)) ↔ (xFz ∧ yFz)))
19	18	anandis 803	. . . . . . . . . . . 12 ⊢ ((F Fn A ∧ (x ∈ A ∧ y ∈ A)) → ((z = (F ‘x) ∧ z = (F ‘y)) ↔ (xFz ∧ yFz)))
20	19	pm5.32da 622	. . . . . . . . . . 11 ⊢ (F Fn A → (((x ∈ A ∧ y ∈ A) ∧ (z = (F ‘x) ∧ z = (F ‘y))) ↔ ((x ∈ A ∧ y ∈ A) ∧ (xFz ∧ yFz))))
21	11, 20	bitr4d 247	. . . . . . . . . 10 ⊢ (F Fn A → ((xFz ∧ yFz) ↔ ((x ∈ A ∧ y ∈ A) ∧ (z = (F ‘x) ∧ z = (F ‘y)))))
22	21	imbi1d 308	. . . . . . . . 9 ⊢ (F Fn A → (((xFz ∧ yFz) → x = y) ↔ (((x ∈ A ∧ y ∈ A) ∧ (z = (F ‘x) ∧ z = (F ‘y))) → x = y)))
23		impexp 433	. . . . . . . . 9 ⊢ ((((x ∈ A ∧ y ∈ A) ∧ (z = (F ‘x) ∧ z = (F ‘y))) → x = y) ↔ ((x ∈ A ∧ y ∈ A) → ((z = (F ‘x) ∧ z = (F ‘y)) → x = y)))
24	22, 23	syl6bb 252	. . . . . . . 8 ⊢ (F Fn A → (((xFz ∧ yFz) → x = y) ↔ ((x ∈ A ∧ y ∈ A) → ((z = (F ‘x) ∧ z = (F ‘y)) → x = y))))
25	24	albidv 1625	. . . . . . 7 ⊢ (F Fn A → (∀z((xFz ∧ yFz) → x = y) ↔ ∀z((x ∈ A ∧ y ∈ A) → ((z = (F ‘x) ∧ z = (F ‘y)) → x = y))))
26		19.21v 1890	. . . . . . . 8 ⊢ (∀z((x ∈ A ∧ y ∈ A) → ((z = (F ‘x) ∧ z = (F ‘y)) → x = y)) ↔ ((x ∈ A ∧ y ∈ A) → ∀z((z = (F ‘x) ∧ z = (F ‘y)) → x = y)))
27		19.23v 1891	. . . . . . . . . 10 ⊢ (∀z((z = (F ‘x) ∧ z = (F ‘y)) → x = y) ↔ (∃z(z = (F ‘x) ∧ z = (F ‘y)) → x = y))
28		fvex 5340	. . . . . . . . . . . 12 ⊢ (F ‘x) ∈ V
29	28	eqvinc 2967	. . . . . . . . . . 11 ⊢ ((F ‘x) = (F ‘y) ↔ ∃z(z = (F ‘x) ∧ z = (F ‘y)))
30	29	imbi1i 315	. . . . . . . . . 10 ⊢ (((F ‘x) = (F ‘y) → x = y) ↔ (∃z(z = (F ‘x) ∧ z = (F ‘y)) → x = y))
31	27, 30	bitr4i 243	. . . . . . . . 9 ⊢ (∀z((z = (F ‘x) ∧ z = (F ‘y)) → x = y) ↔ ((F ‘x) = (F ‘y) → x = y))
32	31	imbi2i 303	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ A) → ∀z((z = (F ‘x) ∧ z = (F ‘y)) → x = y)) ↔ ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
33	26, 32	bitri 240	. . . . . . 7 ⊢ (∀z((x ∈ A ∧ y ∈ A) → ((z = (F ‘x) ∧ z = (F ‘y)) → x = y)) ↔ ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
34	25, 33	syl6bb 252	. . . . . 6 ⊢ (F Fn A → (∀z((xFz ∧ yFz) → x = y) ↔ ((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y))))
35	34	2albidv 1627	. . . . 5 ⊢ (F Fn A → (∀x∀y∀z((xFz ∧ yFz) → x = y) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y))))
36		breq1 4643	. . . . . . . 8 ⊢ (x = y → (xFz ↔ yFz))
37	36	mo4 2237	. . . . . . 7 ⊢ (∃*x xFz ↔ ∀x∀y((xFz ∧ yFz) → x = y))
38	37	albii 1566	. . . . . 6 ⊢ (∀z∃*x xFz ↔ ∀z∀x∀y((xFz ∧ yFz) → x = y))
39		alcom 1737	. . . . . 6 ⊢ (∀z∀x∀y((xFz ∧ yFz) → x = y) ↔ ∀x∀z∀y((xFz ∧ yFz) → x = y))
40		alcom 1737	. . . . . . 7 ⊢ (∀z∀y((xFz ∧ yFz) → x = y) ↔ ∀y∀z((xFz ∧ yFz) → x = y))
41	40	albii 1566	. . . . . 6 ⊢ (∀x∀z∀y((xFz ∧ yFz) → x = y) ↔ ∀x∀y∀z((xFz ∧ yFz) → x = y))
42	38, 39, 41	3bitri 262	. . . . 5 ⊢ (∀z∃*x xFz ↔ ∀x∀y∀z((xFz ∧ yFz) → x = y))
43		r2al 2652	. . . . 5 ⊢ (∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → ((F ‘x) = (F ‘y) → x = y)))
44	35, 42, 43	3bitr4g 279	. . . 4 ⊢ (F Fn A → (∀z∃*x xFz ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
45	2, 44	syl 15	. . 3 ⊢ (F:A–→B → (∀z∃*x xFz ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
46	45	pm5.32i 618	. 2 ⊢ ((F:A–→B ∧ ∀z∃*x xFz) ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
47	1, 46	bitri 240	1 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615   class class class wbr 4640  dom cdm 4773   Fn wfn 4777  –→wf 4778  –1-1→wf1 4779   ‘cfv 4782

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-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fv 4796

This theorem is used by:  dff13f  5473  f1fveq  5474  dff1o6  5476