f11o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > f11o		Structured version Visualization version GIF version

Theorem f11o 7948

Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

**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 groups: 𝑥,𝐹 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem f11o**
Step	Hyp	Ref	Expression
1		f11o.1	. . . 4 ⊢ 𝐹 ∈ V
2	1	ffoss 7947	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
3	2	anbi1i 636	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
4		df-f1 6542	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
5		dff1o3 6828	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝑥 ↔ (𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹))
6	5	anbi1i 636	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵))
7		an32 659	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
8	6, 7	bitri 278	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
9	8	exbii 1881	. . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
10		19.41v 1982	. . 3 ⊢ (∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
11	9, 10	bitri 278	. 2 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
12	3, 4, 11	3bitr4i 306	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∃wex 1812 ∈ wcel 2145 Vcvv 3453 ⊆ wss 3902 ^◡ccnv 5658 Fun wfun 6531 ⟶wf 6533 –1-1→wf1 6534 –onto→wfo 6535 –1-1-onto→wf1o 6536

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2734 ax-sep 5255 ax-pr 5402 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-cnv 5667 df-dm 5669 df-rn 5670 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544

This theorem is used by: domen 8971 uspgrsprfo 49072

W3C validator