Theorem f1oun 5305

Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by set.mm contributors, 26-Mar-1998.)

Assertion

Ref	Expression
f1oun	⊢ (((F:A–1-1-onto→B ∧ G:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (F ∪ G):(A ∪ C)–1-1-onto→(B ∪ D))

Proof of Theorem f1oun

Step	Hyp	Ref	Expression
1		dff1o4 5295	. . . 4 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))
2		dff1o4 5295	. . . 4 ⊢ (G:C–1-1-onto→D ↔ (G Fn C ∧ ◡G Fn D))
3		fnun 5190	. . . . . . 7 ⊢ (((F Fn A ∧ G Fn C) ∧ (A ∩ C) = ∅) → (F ∪ G) Fn (A ∪ C))
4	3	ex 423	. . . . . 6 ⊢ ((F Fn A ∧ G Fn C) → ((A ∩ C) = ∅ → (F ∪ G) Fn (A ∪ C)))
5		fnun 5190	. . . . . . . 8 ⊢ (((◡F Fn B ∧ ◡G Fn D) ∧ (B ∩ D) = ∅) → (◡F ∪ ◡G) Fn (B ∪ D))
6		cnvun 5034	. . . . . . . . 9 ⊢ ◡(F ∪ G) = (◡F ∪ ◡G)
7	6	fneq1i 5179	. . . . . . . 8 ⊢ (◡(F ∪ G) Fn (B ∪ D) ↔ (◡F ∪ ◡G) Fn (B ∪ D))
8	5, 7	sylibr 203	. . . . . . 7 ⊢ (((◡F Fn B ∧ ◡G Fn D) ∧ (B ∩ D) = ∅) → ◡(F ∪ G) Fn (B ∪ D))
9	8	ex 423	. . . . . 6 ⊢ ((◡F Fn B ∧ ◡G Fn D) → ((B ∩ D) = ∅ → ◡(F ∪ G) Fn (B ∪ D)))
10	4, 9	im2anan9 808	. . . . 5 ⊢ (((F Fn A ∧ G Fn C) ∧ (◡F Fn B ∧ ◡G Fn D)) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → ((F ∪ G) Fn (A ∪ C) ∧ ◡(F ∪ G) Fn (B ∪ D))))
11	10	an4s 799	. . . 4 ⊢ (((F Fn A ∧ ◡F Fn B) ∧ (G Fn C ∧ ◡G Fn D)) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → ((F ∪ G) Fn (A ∪ C) ∧ ◡(F ∪ G) Fn (B ∪ D))))
12	1, 2, 11	syl2anb 465	. . 3 ⊢ ((F:A–1-1-onto→B ∧ G:C–1-1-onto→D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → ((F ∪ G) Fn (A ∪ C) ∧ ◡(F ∪ G) Fn (B ∪ D))))
13		dff1o4 5295	. . 3 ⊢ ((F ∪ G):(A ∪ C)–1-1-onto→(B ∪ D) ↔ ((F ∪ G) Fn (A ∪ C) ∧ ◡(F ∪ G) Fn (B ∪ D)))
14	12, 13	syl6ibr 218	. 2 ⊢ ((F:A–1-1-onto→B ∧ G:C–1-1-onto→D) → (((A ∩ C) = ∅ ∧ (B ∩ D) = ∅) → (F ∪ G):(A ∪ C)–1-1-onto→(B ∪ D)))
15	14	imp 418	1 ⊢ (((F:A–1-1-onto→B ∧ G:C–1-1-onto→D) ∧ ((A ∩ C) = ∅ ∧ (B ∩ D) = ∅)) → (F ∪ G):(A ∪ C)–1-1-onto→(B ∪ D))

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  ^◡ccnv 4772   Fn wfn 4777  –1-1-onto→wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  unen  6049