Theorem f1oun 5634

Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Assertion

Ref	Expression
f1oun	|- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) )

Proof of Theorem f1oun

Step	Hyp	Ref	Expression
1		dff1o4 5622	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
2		dff1o4 5622	. . . 4 |- ( G : C -1-1-onto-> D <-> ( G Fn C /\ `' G Fn D ) )
3		fnun 5464	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn C ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
4	3	ex 115	. . . . . 6 |- ( ( F Fn A /\ G Fn C ) -> ( ( A i^i C ) = (/) -> ( F u. G ) Fn ( A u. C ) ) )
5		fnun 5464	. . . . . . . 8 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> ( `' F u. `' G ) Fn ( B u. D ) )
6		cnvun 5168	. . . . . . . . 9 |- `' ( F u. G ) = ( `' F u. `' G )
7	6	fneq1i 5450	. . . . . . . 8 |- ( `' ( F u. G ) Fn ( B u. D ) <-> ( `' F u. `' G ) Fn ( B u. D ) )
8	5, 7	sylibr 134	. . . . . . 7 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> `' ( F u. G ) Fn ( B u. D ) )
9	8	ex 115	. . . . . 6 |- ( ( `' F Fn B /\ `' G Fn D ) -> ( ( B i^i D ) = (/) -> `' ( F u. G ) Fn ( B u. D ) ) )
10	4, 9	im2anan9 602	. . . . 5 |- ( ( ( F Fn A /\ G Fn C ) /\ ( `' F Fn B /\ `' G Fn D ) ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( ( F u. G ) Fn ( A u. C ) /\ `' ( F u. G ) Fn ( B u. D ) ) ) )
11	10	an4s 592	. . . 4 |- ( ( ( F Fn A /\ `' F Fn B ) /\ ( G Fn C /\ `' G Fn D ) ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( ( F u. G ) Fn ( A u. C ) /\ `' ( F u. G ) Fn ( B u. D ) ) ) )
12	1, 2, 11	syl2anb 291	. . 3 |- ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( ( F u. G ) Fn ( A u. C ) /\ `' ( F u. G ) Fn ( B u. D ) ) ) )
13		dff1o4 5622	. . 3 |- ( ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) <-> ( ( F u. G ) Fn ( A u. C ) /\ `' ( F u. G ) Fn ( B u. D ) ) )
14	12, 13	imbitrrdi 162	. 2 |- ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) -> ( ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) ) )
15	14	imp 124	1 |- ( ( ( F : A -1-1-onto-> B /\ G : C -1-1-onto-> D ) /\ ( ( A i^i C ) = (/) /\ ( B i^i D ) = (/) ) ) -> ( F u. G ) : ( A u. C ) -1-1-onto-> ( B u. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   u. cun 3209   i^i cin 3210   (/)c0 3508   `'ccnv 4748   Fn wfn 5347   -1-1-onto->wf1o 5351

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by:  f1oprg  5660  unen  7058  zfz1isolem1  11212  ennnfonelemhf1o  13164  gfsump1  16868