Theorem fun11 5160

Description: Two ways of stating that A is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un₂ (A)" for one-to-one. (Contributed by NM, 17-Jan-2006.) (Revised by Scott Fenton, 18-Apr-2021.)

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 609	. . . . . . . 8 ⊢ ((x = z ↔ y = w) ↔ ((x = z → y = w) ∧ (y = w → x = z)))
2	1	imbi2i 303	. . . . . . 7 ⊢ (((xAy ∧ zAw) → (x = z ↔ y = w)) ↔ ((xAy ∧ zAw) → ((x = z → y = w) ∧ (y = w → x = z))))
3		pm4.76 836	. . . . . . 7 ⊢ ((((xAy ∧ zAw) → (x = z → y = w)) ∧ ((xAy ∧ zAw) → (y = w → x = z))) ↔ ((xAy ∧ zAw) → ((x = z → y = w) ∧ (y = w → x = z))))
4		bi2.04 350	. . . . . . . 8 ⊢ (((xAy ∧ zAw) → (x = z → y = w)) ↔ (x = z → ((xAy ∧ zAw) → y = w)))
5		bi2.04 350	. . . . . . . 8 ⊢ (((xAy ∧ zAw) → (y = w → x = z)) ↔ (y = w → ((xAy ∧ zAw) → x = z)))
6	4, 5	anbi12i 678	. . . . . . 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 264	. . . . . 6 ⊢ (((xAy ∧ zAw) → (x = z ↔ y = w)) ↔ ((x = z → ((xAy ∧ zAw) → y = w)) ∧ (y = w → ((xAy ∧ zAw) → x = z))))
8	7	2albii 1567	. . . . 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 1594	. . . . 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 1737	. . . . . . 7 ⊢ (∀x∀y(x = z → ((xAy ∧ zAw) → y = w)) ↔ ∀y∀x(x = z → ((xAy ∧ zAw) → y = w)))
11		nfv 1619	. . . . . . . . 9 ⊢ Ⅎx((zAy ∧ zAw) → y = w)
12		breq1 4643	. . . . . . . . . . 11 ⊢ (x = z → (xAy ↔ zAy))
13	12	anbi1d 685	. . . . . . . . . 10 ⊢ (x = z → ((xAy ∧ zAw) ↔ (zAy ∧ zAw)))
14	13	imbi1d 308	. . . . . . . . 9 ⊢ (x = z → (((xAy ∧ zAw) → y = w) ↔ ((zAy ∧ zAw) → y = w)))
15	11, 14	equsal 1960	. . . . . . . 8 ⊢ (∀x(x = z → ((xAy ∧ zAw) → y = w)) ↔ ((zAy ∧ zAw) → y = w))
16	15	albii 1566	. . . . . . 7 ⊢ (∀y∀x(x = z → ((xAy ∧ zAw) → y = w)) ↔ ∀y((zAy ∧ zAw) → y = w))
17	10, 16	bitri 240	. . . . . 6 ⊢ (∀x∀y(x = z → ((xAy ∧ zAw) → y = w)) ↔ ∀y((zAy ∧ zAw) → y = w))
18		nfv 1619	. . . . . . . 8 ⊢ Ⅎy((xAw ∧ zAw) → x = z)
19		breq2 4644	. . . . . . . . . 10 ⊢ (y = w → (xAy ↔ xAw))
20	19	anbi1d 685	. . . . . . . . 9 ⊢ (y = w → ((xAy ∧ zAw) ↔ (xAw ∧ zAw)))
21	20	imbi1d 308	. . . . . . . 8 ⊢ (y = w → (((xAy ∧ zAw) → x = z) ↔ ((xAw ∧ zAw) → x = z)))
22	18, 21	equsal 1960	. . . . . . 7 ⊢ (∀y(y = w → ((xAy ∧ zAw) → x = z)) ↔ ((xAw ∧ zAw) → x = z))
23	22	albii 1566	. . . . . 6 ⊢ (∀x∀y(y = w → ((xAy ∧ zAw) → x = z)) ↔ ∀x((xAw ∧ zAw) → x = z))
24	17, 23	anbi12i 678	. . . . 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)))
25	8, 9, 24	3bitri 262	. . . 4 ⊢ (∀x∀y((xAy ∧ zAw) → (x = z ↔ y = w)) ↔ (∀y((zAy ∧ zAw) → y = w) ∧ ∀x((xAw ∧ zAw) → x = z)))
26	25	2albii 1567	. . 3 ⊢ (∀z∀w∀x∀y((xAy ∧ zAw) → (x = z ↔ y = w)) ↔ ∀z∀w(∀y((zAy ∧ zAw) → y = w) ∧ ∀x((xAw ∧ zAw) → x = z)))
27		19.26-2 1594	. . 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)))
28	26, 27	bitr2i 241	. 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)))
29		dffun2 5120	. . . 4 ⊢ (Fun A ↔ ∀z∀y∀w((zAy ∧ zAw) → y = w))
30		alcom 1737	. . . . 5 ⊢ (∀y∀w((zAy ∧ zAw) → y = w) ↔ ∀w∀y((zAy ∧ zAw) → y = w))
31	30	albii 1566	. . . 4 ⊢ (∀z∀y∀w((zAy ∧ zAw) → y = w) ↔ ∀z∀w∀y((zAy ∧ zAw) → y = w))
32	29, 31	bitri 240	. . 3 ⊢ (Fun A ↔ ∀z∀w∀y((zAy ∧ zAw) → y = w))
33		brcnv 4893	. . . . . . . 8 ⊢ (w◡Ax ↔ xAw)
34		brcnv 4893	. . . . . . . 8 ⊢ (w◡Az ↔ zAw)
35	33, 34	anbi12i 678	. . . . . . 7 ⊢ ((w◡Ax ∧ w◡Az) ↔ (xAw ∧ zAw))
36	35	imbi1i 315	. . . . . 6 ⊢ (((w◡Ax ∧ w◡Az) → x = z) ↔ ((xAw ∧ zAw) → x = z))
37	36	albii 1566	. . . . 5 ⊢ (∀z((w◡Ax ∧ w◡Az) → x = z) ↔ ∀z((xAw ∧ zAw) → x = z))
38	37	2albii 1567	. . . 4 ⊢ (∀w∀x∀z((w◡Ax ∧ w◡Az) → x = z) ↔ ∀w∀x∀z((xAw ∧ zAw) → x = z))
39		dffun2 5120	. . . 4 ⊢ (Fun ◡A ↔ ∀w∀x∀z((w◡Ax ∧ w◡Az) → x = z))
40		alrot3 1738	. . . 4 ⊢ (∀z∀w∀x((xAw ∧ zAw) → x = z) ↔ ∀w∀x∀z((xAw ∧ zAw) → x = z))
41	38, 39, 40	3bitr4i 268	. . 3 ⊢ (Fun ◡A ↔ ∀z∀w∀x((xAw ∧ zAw) → x = z))
42	32, 41	anbi12i 678	. 2 ⊢ ((Fun A ∧ Fun ◡A) ↔ (∀z∀w∀y((zAy ∧ zAw) → y = w) ∧ ∀z∀w∀x((xAw ∧ zAw) → x = z)))
43		alrot4 1739	. 2 ⊢ (∀x∀y∀z∀w((xAy ∧ zAw) → (x = z ↔ y = w)) ↔ ∀z∀w∀x∀y((xAy ∧ zAw) → (x = z ↔ y = w)))
44	28, 42, 43	3bitr4i 268	1 ⊢ ((Fun A ∧ Fun ◡A) ↔ ∀x∀y∀z∀w((xAy ∧ zAw) → (x = z ↔ y = w)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   class class class wbr 4640  ^◡ccnv 4772  Fun wfun 4776

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 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 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 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-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by: (None)