Theorem f1oun 2815

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

Proof of Theorem f1oun

Step	Hyp	Ref	Expression
1		fnun 2730	. . . . . . 7 ⊢ (((F Fn A ∧ G Fn C) ∧ (A ∩ C) = ∅) → (F ∪ G) Fn (A ∪ C))
2	1	exp 291	. . . . . 6 ⊢ ((F Fn A ∧ G Fn C) → ((A ∩ C) = ∅ → (F ∪ G) Fn (A ∪ C)))
3		fnun 2730	. . . . . . . 8 ⊢ (((◡F Fn B ∧ ◡G Fn D) ∧ (B ∩ D) = ∅) → (◡F ∪ ◡G) Fn (B ∪ D))
4		cnvun 2642	. . . . . . . . 9 ⊢ ◡(F ∪ G) = (◡F ∪ ◡G)
5		fneq1 2718	. . . . . . . . 9 ⊢ (◡(F ∪ G) = (◡F ∪ ◡G) → (◡(F ∪ G) Fn (B ∪ D) ↔ (◡F ∪ ◡G) Fn (B ∪ D)))
6	4, 5	ax-mp 6	. . . . . . . 8 ⊢ (◡(F ∪ G) Fn (B ∪ D) ↔ (◡F ∪ ◡G) Fn (B ∪ D))
7	3, 6	sylibr 175	. . . . . . 7 ⊢ (((◡F Fn B ∧ ◡G Fn D) ∧ (B ∩ D) = ∅) → ◡(F ∪ G) Fn (B ∪ D))
8	7	exp 291	. . . . . 6 ⊢ ((◡F Fn B ∧ ◡G Fn D) → ((B ∩ D) = ∅ → ◡(F ∪ G) Fn (B ∪ D)))
9	2, 8	im2anan9 434	. . . . 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))))
10	9	an4s 390	. . . 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))))
11		f1o4 2807	. . . 4 ⊢ (F:A–1-1-onto→B ↔ (F Fn A ∧ ◡F Fn B))
12		f1o4 2807	. . . 4 ⊢ (G:C–1-1-onto→D ↔ (G Fn C ∧ ◡G Fn D))
13	10, 11, 12	syl2anb 350	. . 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))))
14		f1o4 2807	. . 3 ⊢ ((F ∪ G):(A ∪ C)–1-1-onto→(B ∪ D) ↔ ((F ∪ G) Fn (A ∪ C) ∧ ◡(F ∪ G) Fn (B ∪ D)))
15	13, 14	syl6ibr 186	. 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)))
16	15	imp 277	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 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  ^◡ccnv 2409   Fn wfn 2417  –1-1-onto→wf1o 2421

This theorem is referenced by:  unen 3338  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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