Theorem f1ocnv2d 5974

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 2211	. . . . . 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 376	. . . 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 295	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
9	8	exp42 368	. . . . . 6 |- ( ph -> ( x e. A -> ( y e. B -> ( y = C -> x = D ) ) ) )
10	9	com34 83	. . . . 5 |- ( ph -> ( x e. A -> ( y = C -> ( y e. B -> x = D ) ) ) )
11	10	imp32 255	. . . 4 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
12	6, 11	jcai 309	. . 3 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
13		eleq1a 2211	. . . . . 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 376	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
16	7	biimpa 294	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
17	16	exp42 368	. . . . . . 7 |- ( ph -> ( x e. A -> ( y e. B -> ( x = D -> y = C ) ) ) )
18	17	com23 78	. . . . . 6 |- ( ph -> ( y e. B -> ( x e. A -> ( x = D -> y = C ) ) ) )
19	18	com34 83	. . . . 5 |- ( ph -> ( y e. B -> ( x = D -> ( x e. A -> y = C ) ) ) )
20	19	imp32 255	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
21	15, 20	jcai 309	. . 3 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
22	12, 21	impbida 585	. 2 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
23	1, 2, 3, 22	f1ocnvd 5972	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   |-> cmpt 3989   `'ccnv 4538   -1-1-onto->wf1o 5122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130

This theorem is referenced by:  f1o2d  5975  negf1o  8144  negiso  8713  iccf1o  9787  xrnegiso  11031  txhmeo  12488