Theorem f1ocnv2d 6088

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 2259	. . . . . 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 379	. . . 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 297	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
9	8	exp42 371	. . . . . 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 257	. . . 4 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
12	6, 11	jcai 311	. . 3 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
13		eleq1a 2259	. . . . . 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 379	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
16	7	biimpa 296	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
17	16	exp42 371	. . . . . . 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 257	. . . 4 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
21	15, 20	jcai 311	. . 3 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
22	12, 21	impbida 596	. 2 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
23	1, 2, 3, 22	f1ocnvd 6086	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   |-> cmpt 4076   `'ccnv 4637   -1-1-onto->wf1o 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235

This theorem is referenced by:  f1o2d  6089  negf1o  8352  negiso  8925  iccf1o  10017  xrnegiso  11283  grpinvcnv  12962  grplactcnv  12996  txhmeo  14059