Theorem f1oun 5612

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 5600	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
2		dff1o4 5600	. . . 4 |- ( G : C -1-1-onto-> D <-> ( G Fn C /\ `' G Fn D ) )
3		fnun 5445	. . . . . . 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 5445	. . . . . . . 8 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> ( `' F u. `' G ) Fn ( B u. D ) )
6		cnvun 5149	. . . . . . . . 9 |- `' ( F u. G ) = ( `' F u. `' G )
7	6	fneq1i 5431	. . . . . . . 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 5600	. . 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 3199   i^i cin 3200   (/)c0 3496   `'ccnv 4730   Fn wfn 5328   -1-1-onto->wf1o 5332

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340

This theorem is referenced by:  f1oprg  5638  unen  7034  zfz1isolem1  11167  ennnfonelemhf1o  13114  gfsump1  16815