Theorem f1oun 3697

Description: The union of two one-to-one onto functions with disjoint domains and ranges.

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		fnun 3586	. . . . . . 7 |- (((F Fn A /\ G Fn C) /\ (A i^i C) = (/)) -> (F u. G) Fn (A u. C))
2	1	ex 373	. . . . . 6 |- ((F Fn A /\ G Fn C) -> ((A i^i C) = (/) -> (F u. G) Fn (A u. C)))
3		fnun 3586	. . . . . . . 8 |- (((`'F Fn B /\ `'G Fn D) /\ (B i^i D) = (/)) -> (`'F u. `'G) Fn (B u. D))
4		cnvun 3447	. . . . . . . . 9 |- `'(F u. G) = (`'F u. `'G)
5		fneq1 3574	. . . . . . . . 9 |- (`'(F u. G) = (`'F u. `'G) -> (`'(F u. G) Fn (B u. D) <-> (`'F u. `'G) Fn (B u. D)))
6	4, 5	ax-mp 7	. . . . . . . 8 |- (`'(F u. G) Fn (B u. D) <-> (`'F u. `'G) Fn (B u. D))
7	3, 6	sylibr 200	. . . . . . 7 |- (((`'F Fn B /\ `'G Fn D) /\ (B i^i D) = (/)) -> `'(F u. G) Fn (B u. D))
8	7	ex 373	. . . . . 6 |- ((`'F Fn B /\ `'G Fn D) -> ((B i^i D) = (/) -> `'(F u. G) Fn (B u. D)))
9	2, 8	im2anan9 562	. . . . 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))))
10	9	an4s 508	. . . 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))))
11		f1o4 3687	. . . 4 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
12		f1o4 3687	. . . 4 |- (G:C-1-1-onto->D <-> (G Fn C /\ `'G Fn D))
13	10, 11, 12	syl2anb 455	. . 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))))
14		f1o4 3687	. . 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)))
15	13, 14	syl6ibr 213	. 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)))
16	15	imp 350	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 3   <-> wb 146   /\ wa 223   = wceq 954   u. cun 2041   i^i cin 2042  (/)c0 2276  `'ccnv 3164   Fn wfn 3172  -1-1-onto->wf1o 3176

This theorem is referenced by:  unen 4420  infxpidmlem11 7513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192

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