Theorem f1ocnv2d 5807

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

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
f1ocnv2d	|- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

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 f1ocnv2d

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 2156	. . . . . 6 |- ( C e. B -> ( y = C -> y e. B ) )
5	2, 4	syl 14	. . . . 5 |- ( ( ph /\ x e. A ) -> ( y = C -> y e. B ) )
6	5	impr 371	. . . 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 291	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
9	8	exp42 363	. . . . . 6 |- ( ph -> ( x e. A -> ( y e. B -> ( y = C -> x = D ) ) ) )
10	9	com34 82	. . . . 5 |- ( ph -> ( x e. A -> ( y = C -> ( y e. B -> x = D ) ) ) )
11	10	imp32 253	. . . 4 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
12	6, 11	jcai 304	. . 3 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
13		eleq1a 2156	. . . . . 6 |- ( D e. A -> ( x = D -> x e. A ) )
14	3, 13	syl 14	. . . . 5 |- ( ( ph /\ y e. B ) -> ( x = D -> x e. A ) )
15	14	impr 371	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
16	7	biimpa 290	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
17	16	exp42 363	. . . . . . 7 |- ( ph -> ( x e. A -> ( y e. B -> ( x = D -> y = C ) ) ) )
18	17	com23 77	. . . . . 6 |- ( ph -> ( y e. B -> ( x e. A -> ( x = D -> y = C ) ) ) )
19	18	com34 82	. . . . 5 |- ( ph -> ( y e. B -> ( x = D -> ( x e. A -> y = C ) ) ) )
20	19	imp32 253	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
21	15, 20	jcai 304	. . 3 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
22	12, 21	impbida 561	. 2 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
23	1, 2, 3, 22	f1ocnvd 5805	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1287   e. wcel 1436   |-> cmpt 3876   `'ccnv 4412   -1-1-onto->wf1o 4982

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990

This theorem is referenced by:  f1o2d  5808  negf1o  7807  negiso  8354  iccf1o  9356