f11o - Metamath Proof Explorer

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Theorem f11o 7971

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 7970	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
3	2	anbi1i 624	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
4		df-f1 6566	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
5		dff1o3 6854	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝑥 ↔ (𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹))
6	5	anbi1i 624	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵))
7		an32 646	. . . . 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 1848	. . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
10		19.41v 1949	. . 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 1779 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ^◡ccnv 5684 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558 –onto→wfo 6559 –1-1-onto→wf1o 6560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568

This theorem is referenced by: domen 9002 uspgrsprfo 48064

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