Theorem f1oun 3782

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 3669	. . . . . . 7 |- (((F Fn A /\ G Fn C) /\ (A i^i C) = (/)) -> (F u. G) Fn (A u. C))
2	1	ex 371	. . . . . 6 |- ((F Fn A /\ G Fn C) -> ((A i^i C) = (/) -> (F u. G) Fn (A u. C)))
3		fnun 3669	. . . . . . . 8 |- (((`'F Fn B /\ `'G Fn D) /\ (B i^i D) = (/)) -> (`'F u. `'G) Fn (B u. D))
4		cnvun 3511	. . . . . . . . 9 |- `'(F u. G) = (`'F u. `'G)
5		fneq1 3657	. . . . . . . . 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 198	. . . . . . 7 |- (((`'F Fn B /\ `'G Fn D) /\ (B i^i D) = (/)) -> `'(F u. G) Fn (B u. D))
8	7	ex 371	. . . . . 6 |- ((`'F Fn B /\ `'G Fn D) -> ((B i^i D) = (/) -> `'(F u. G) Fn (B u. D)))
9	2, 8	im2anan9 565	. . . . 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 510	. . . 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		dff1o4 3772	. . . 4 |- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
12		dff1o4 3772	. . . 4 |- (G:C-1-1-onto->D <-> (G Fn C /\ `'G Fn D))
13	10, 11, 12	syl2anb 457	. . 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		dff1o4 3772	. . 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 211	. 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 348	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 144   /\ wa 221   = wceq 988   u. cun 2089   i^i cin 2090  (/)c0 2324  `'ccnv 3224   Fn wfn 3232  -1-1-onto->wf1o 3236

This theorem is referenced by:  unen 4521  infxpidmlem11 7687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252

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