Theorem f1o2d 5728

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

Hypotheses

Ref	Expression
f1od.1	|- F = (x e. A |-> C)
f1o2d.2	|- ((ph /\ x e. A) -> C e. B)
f1o2d.3	|- ((ph /\ y e. B) -> D e. A)
f1o2d.4	|- ((ph /\ (x e. A /\ y e. B)) -> (x = D <-> y = C))

Assertion

Ref	Expression
f1o2d	|- (ph -> F:A-1-1-onto->B)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D   ph,x,y

Allowed substitution hints:   C(x)   D(y)   F(x,y)

Proof of Theorem f1o2d

Step	Hyp	Ref	Expression
1		f1od.1	. 2 |- F = (x e. A |-> C)
2		f1o2d.2	. 2 |- ((ph /\ x e. A) -> C e. B)
3		f1o2d.3	. 2 |- ((ph /\ y e. B) -> D e. A)
4		eleq1a 2422	. . . . . 6 |- (C e. B -> (y = C -> y e. B))
5	2, 4	syl 15	. . . . 5 |- ((ph /\ x e. A) -> (y = C -> y e. B))
6	5	impr 602	. . . 4 |- ((ph /\ (x e. A /\ y = C)) -> y e. B)
7		f1o2d.4	. . . . . . . 8 |- ((ph /\ (x e. A /\ y e. B)) -> (x = D <-> y = C))
8	7	biimpar 471	. . . . . . 7 |- (((ph /\ (x e. A /\ y e. B)) /\ y = C) -> x = D)
9	8	exp42 594	. . . . . 6 |- (ph -> (x e. A -> (y e. B -> (y = C -> x = D))))
10	9	com34 77	. . . . 5 |- (ph -> (x e. A -> (y = C -> (y e. B -> x = D))))
11	10	imp32 422	. . . 4 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B -> x = D))
12	6, 11	jcai 522	. . 3 |- ((ph /\ (x e. A /\ y = C)) -> (y e. B /\ x = D))
13		eleq1a 2422	. . . . . 6 |- (D e. A -> (x = D -> x e. A))
14	3, 13	syl 15	. . . . 5 |- ((ph /\ y e. B) -> (x = D -> x e. A))
15	14	impr 602	. . . 4 |- ((ph /\ (y e. B /\ x = D)) -> x e. A)
16	7	biimpa 470	. . . . . . . 8 |- (((ph /\ (x e. A /\ y e. B)) /\ x = D) -> y = C)
17	16	exp42 594	. . . . . . 7 |- (ph -> (x e. A -> (y e. B -> (x = D -> y = C))))
18	17	com23 72	. . . . . 6 |- (ph -> (y e. B -> (x e. A -> (x = D -> y = C))))
19	18	com34 77	. . . . 5 |- (ph -> (y e. B -> (x = D -> (x e. A -> y = C))))
20	19	imp32 422	. . . 4 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A -> y = C))
21	15, 20	jcai 522	. . 3 |- ((ph /\ (y e. B /\ x = D)) -> (x e. A /\ y = C))
22	12, 21	impbida 805	. 2 |- (ph -> ((x e. A /\ y = C) <-> (y e. B /\ x = D)))
23	1, 2, 3, 22	f1od 5727	1 |- (ph -> F:A-1-1-onto->B)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  -1-1-onto->wf1o 4781   |-> cmpt 5652

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-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-fo 4794  df-f1o 4795  df-mpt 5653

This theorem is referenced by: (None)