Theorem f1oun 5520

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 5508	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
2		dff1o4 5508	. . . 4 |- ( G : C -1-1-onto-> D <-> ( G Fn C /\ `' G Fn D ) )
3		fnun 5360	. . . . . . 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 5360	. . . . . . . 8 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> ( `' F u. `' G ) Fn ( B u. D ) )
6		cnvun 5071	. . . . . . . . 9 |- `' ( F u. G ) = ( `' F u. `' G )
7	6	fneq1i 5348	. . . . . . . 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 598	. . . . 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 588	. . . 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 5508	. . 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 1364   u. cun 3151   i^i cin 3152   (/)c0 3446   `'ccnv 4658   Fn wfn 5249   -1-1-onto->wf1o 5253

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261

This theorem is referenced by:  f1oprg  5544  unen  6870  zfz1isolem1  10911  ennnfonelemhf1o  12570