f11o - Metamath Proof Explorer

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Theorem f11o 7986

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⟶1-1𝐵 ↔ ∃𝑥(𝐹:𝐴⟶1-1onto𝑥 ∧ 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 7985	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵))
3	2	anbi1i 636	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
4		df-f1 6566	. 2 ⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
5		dff1o3 6856	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1onto𝑥 ↔ (𝐹:𝐴⟶onto𝑥 ∧ Fun ^◡𝐹))
6	5	anbi1i 636	. . . . 5 ⊢ ((𝐹:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴⟶onto𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵))
7		an32 659	. . . . 5 ⊢ (((𝐹:𝐴⟶onto𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
8	6, 7	bitri 278	. . . 4 ⊢ ((𝐹:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
9	8	exbii 1881	. . 3 ⊢ (∃𝑥(𝐹:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
10		19.41v 1984	. . 3 ⊢ (∃𝑥((𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
11	9, 10	bitri 278	. 2 ⊢ (∃𝑥(𝐹:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑥(𝐹:𝐴⟶onto𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
12	3, 4, 11	3bitr4i 306	1 ⊢ (𝐹:𝐴⟶1-1𝐵 ↔ ∃𝑥(𝐹:𝐴⟶1-1onto𝑥 ∧ 𝑥 ⊆ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∃wex 1812 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 ^◡ccnv 5655 Fun wfun 6555 ⟶wf 6557 ⟶1-1wf1 6558 ⟶ontowfo 6559 ⟶1-1ontowf1o 6560

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5253 ax-pr 5396 ax-un 7778

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 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-cnv 5664 df-dm 5666 df-rn 5667 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568

This theorem is used by: domen 9019 uspgrsprfo 49461

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