Theorem f1oun 5462

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 5450	. . . 4 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
2		dff1o4 5450	. . . 4 |- ( G : C -1-1-onto-> D <-> ( G Fn C /\ `' G Fn D ) )
3		fnun 5304	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn C ) /\ ( A i^i C ) = (/) ) -> ( F u. G ) Fn ( A u. C ) )
4	3	ex 114	. . . . . 6 |- ( ( F Fn A /\ G Fn C ) -> ( ( A i^i C ) = (/) -> ( F u. G ) Fn ( A u. C ) ) )
5		fnun 5304	. . . . . . . 8 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> ( `' F u. `' G ) Fn ( B u. D ) )
6		cnvun 5016	. . . . . . . . 9 |- `' ( F u. G ) = ( `' F u. `' G )
7	6	fneq1i 5292	. . . . . . . 8 |- ( `' ( F u. G ) Fn ( B u. D ) <-> ( `' F u. `' G ) Fn ( B u. D ) )
8	5, 7	sylibr 133	. . . . . . 7 |- ( ( ( `' F Fn B /\ `' G Fn D ) /\ ( B i^i D ) = (/) ) -> `' ( F u. G ) Fn ( B u. D ) )
9	8	ex 114	. . . . . 6 |- ( ( `' F Fn B /\ `' G Fn D ) -> ( ( B i^i D ) = (/) -> `' ( F u. G ) Fn ( B u. D ) ) )
10	4, 9	im2anan9 593	. . . . 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 583	. . . 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 289	. . 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 5450	. . 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	syl6ibr 161	. 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 123	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 103   = wceq 1348   u. cun 3119   i^i cin 3120   (/)c0 3414   `'ccnv 4610   Fn wfn 5193   -1-1-onto->wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1oprg  5486  unen  6794  zfz1isolem1  10775  ennnfonelemhf1o  12368