Theorem f11o 2821

Description: Relationship between one-to-one and one-to-one onto function.

Hypothesis

Ref	Expression
f11o.1	⊢ F ∈ V

Assertion

Ref	Expression
f11o	⊢ (F:A–1-1→B ↔ ∃x(F:A–1-1-onto→x ∧ x ⊆ B))

Distinct variable group(s):   x,F   x,A   x,B

Proof of Theorem f11o

Step	Hyp	Ref	Expression
1		f11o.1	. . . 4 ⊢ F ∈ V
2	1	ffoss 2820	. . 3 ⊢ (F:A–→B ↔ ∃x(F:A–onto→x ∧ x ⊆ B))
3	2	anbi1i 368	. 2 ⊢ ((F:A–→B ∧ Fun ◡F) ↔ (∃x(F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
4		df-f1 2435	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
5		f1o3 2805	. . . . . 6 ⊢ (F:A–1-1-onto→x ↔ (F:A–onto→x ∧ Fun ◡F))
6	5	anbi1i 368	. . . . 5 ⊢ ((F:A–1-1-onto→x ∧ x ⊆ B) ↔ ((F:A–onto→x ∧ Fun ◡F) ∧ x ⊆ B))
7		an23 371	. . . . 5 ⊢ (((F:A–onto→x ∧ Fun ◡F) ∧ x ⊆ B) ↔ ((F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
8	6, 7	bitr 151	. . . 4 ⊢ ((F:A–1-1-onto→x ∧ x ⊆ B) ↔ ((F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
9	8	biex 733	. . 3 ⊢ (∃x(F:A–1-1-onto→x ∧ x ⊆ B) ↔ ∃x((F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
10		19.41v 963	. . 3 ⊢ (∃x((F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F) ↔ (∃x(F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
11	9, 10	bitr 151	. 2 ⊢ (∃x(F:A–1-1-onto→x ∧ x ⊆ B) ↔ (∃x(F:A–onto→x ∧ x ⊆ B) ∧ Fun ◡F))
12	3, 4, 11	3bitr4 158	1 ⊢ (F:A–1-1→B ↔ ∃x(F:A–1-1-onto→x ∧ x ⊆ B))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ^◡ccnv 2409  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  domen 3284

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  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-uni 1920  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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