f11o - Metamath Proof Explorer

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Theorem f11o 7893

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 7892	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
3	2	anbi1i 625	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
4		df-f1 6497	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
5		dff1o3 6780	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝑥 ↔ (𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹))
6	5	anbi1i 625	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵))
7		an32 647	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
8	6, 7	bitri 275	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
9	8	exbii 1850	. . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
10		19.41v 1951	. . 3 ⊢ (∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
11	9, 10	bitri 275	. 2 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
12	3, 4, 11	3bitr4i 303	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1781 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 ^◡ccnv 5623 Fun wfun 6486 ⟶wf 6488 –1-1→wf1 6489 –onto→wfo 6490 –1-1-onto→wf1o 6491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5231 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-cnv 5632 df-dm 5634 df-rn 5635 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499

This theorem is referenced by: domen 8901 uspgrsprfo 48636

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