Theorem fun11 5963

Description: Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
fun11	⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴

Proof of Theorem fun11

Step	Hyp	Ref	Expression
1		dfbi2 660	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ↔ 𝑦 = 𝑤) ↔ ((𝑥 = 𝑧 → 𝑦 = 𝑤) ∧ (𝑦 = 𝑤 → 𝑥 = 𝑧)))
2	1	imbi2i 326	. . . . . . 7 ⊢ (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → ((𝑥 = 𝑧 → 𝑦 = 𝑤) ∧ (𝑦 = 𝑤 → 𝑥 = 𝑧))))
3		pm4.76 910	. . . . . . 7 ⊢ ((((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 → 𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑦 = 𝑤 → 𝑥 = 𝑧))) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → ((𝑥 = 𝑧 → 𝑦 = 𝑤) ∧ (𝑦 = 𝑤 → 𝑥 = 𝑧))))
4		bi2.04 376	. . . . . . . 8 ⊢ (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 → 𝑦 = 𝑤)) ↔ (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)))
5		bi2.04 376	. . . . . . . 8 ⊢ (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑦 = 𝑤 → 𝑥 = 𝑧)) ↔ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
6	4, 5	anbi12i 733	. . . . . . 7 ⊢ ((((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 → 𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑦 = 𝑤 → 𝑥 = 𝑧))) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))))
7	2, 3, 6	3bitr2i 288	. . . . . 6 ⊢ (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))))
8	7	2albii 1748	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))))
9		19.26-2 1799	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑥∀𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))))
10		alcom 2037	. . . . . . 7 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)))
11		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥𝐴𝑦 ↔ 𝑧𝐴𝑦))
12	11	anbi1d 741	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) ↔ (𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤)))
13	12	imbi1d 331	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)))
14	13	equsalvw 1931	. . . . . . . 8 ⊢ (∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
15	14	albii 1747	. . . . . . 7 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
16	10, 15	bitri 264	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
17		breq2 4657	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑥𝐴𝑦 ↔ 𝑥𝐴𝑤))
18	17	anbi1d 741	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) ↔ (𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤)))
19	18	imbi1d 331	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
20	19	equsalvw 1931	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
21	20	albii 1747	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
22	16, 21	anbi12i 733	. . . . 5 ⊢ ((∀𝑥∀𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
23	8, 9, 22	3bitri 286	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ (∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
24	23	2albii 1748	. . 3 ⊢ (∀𝑧∀𝑤∀𝑥∀𝑦((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤(∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
25		19.26-2 1799	. . 3 ⊢ (∀𝑧∀𝑤(∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ (∀𝑧∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
26	24, 25	bitr2i 265	. 2 ⊢ ((∀𝑧∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑧∀𝑤∀𝑥∀𝑦((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))
27		fun2cnv 5960	. . . 4 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑧∃*𝑦 𝑧𝐴𝑦)
28		breq2 4657	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧𝐴𝑦 ↔ 𝑧𝐴𝑤))
29	28	mo4 2517	. . . . 5 ⊢ (∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑦∀𝑤((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
30	29	albii 1747	. . . 4 ⊢ (∀𝑧∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑧∀𝑦∀𝑤((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
31		alcom 2037	. . . . 5 ⊢ (∀𝑦∀𝑤((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
32	31	albii 1747	. . . 4 ⊢ (∀𝑧∀𝑦∀𝑤((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑧∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
33	27, 30, 32	3bitri 286	. . 3 ⊢ (Fun ◡◡𝐴 ↔ ∀𝑧∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤))
34		funcnv2 5957	. . . 4 ⊢ (Fun ◡𝐴 ↔ ∀𝑤∃*𝑥 𝑥𝐴𝑤)
35		breq1 4656	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥𝐴𝑤 ↔ 𝑧𝐴𝑤))
36	35	mo4 2517	. . . . 5 ⊢ (∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑥∀𝑧((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
37	36	albii 1747	. . . 4 ⊢ (∀𝑤∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑤∀𝑥∀𝑧((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
38		alcom 2037	. . . . . 6 ⊢ (∀𝑥∀𝑧((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
39	38	albii 1747	. . . . 5 ⊢ (∀𝑤∀𝑥∀𝑧((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑤∀𝑧∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
40		alcom 2037	. . . . 5 ⊢ (∀𝑤∀𝑧∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
41	39, 40	bitri 264	. . . 4 ⊢ (∀𝑤∀𝑥∀𝑧((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
42	34, 37, 41	3bitri 286	. . 3 ⊢ (Fun ◡𝐴 ↔ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧))
43	33, 42	anbi12i 733	. 2 ⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ (∀𝑧∀𝑤∀𝑦((𝑧𝐴𝑦 ∧ 𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧∀𝑤∀𝑥((𝑥𝐴𝑤 ∧ 𝑧𝐴𝑤) → 𝑥 = 𝑧)))
44		alrot4 2039	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤∀𝑥∀𝑦((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))
45	26, 43, 44	3bitr4i 292	1 ⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃*wmo 2471   class class class wbr 4653  ^◡ccnv 5113  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by: (None)