Theorem f1ocnv2d 6065

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 2247	. . . . . 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 2247	. . . . . 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 6063	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   |-> cmpt 4059   `'ccnv 4619   -1-1-onto->wf1o 5207

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215

This theorem is referenced by:  f1o2d  6066  negf1o  8313  negiso  8883  iccf1o  9973  xrnegiso  11236  grpinvcnv  12797  grplactcnv  12831  txhmeo  13388