Theorem f1oun 5304

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 5294	. . . 4 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))
2		dff1o4 5294	. . . 4 ⊢ (G:C–1-1-onto→D ↔ (G Fn C ∧ ◡G Fn D))
3		fnun 5189	. . . . . . 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 5189	. . . . . . . 8 ⊢ (((◡F Fn B ∧ ◡G Fn D) ∧ (B ∩ D) = ∅) → (◡F ∪ ◡G) Fn (B ∪ D))
6		cnvun 5033	. . . . . . . . 9 ⊢ ◡(F ∪ G) = (◡F ∪ ◡G)
7	6	fneq1i 5178	. . . . . . . 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 5294	. . 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 3207   ∩ cin 3208  ∅c0 3550  ^◡ccnv 4771   Fn wfn 4776  –1-1-onto→wf1o 4780

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  unen  6048