f11o - Intuitionistic Logic Explorer

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Theorem f11o 5179

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 5178	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
3	2	anbi1i 445	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
4		df-f1 4927	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
5		dff1o3 5152	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝑥 ↔ (𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹))
6	5	anbi1i 445	. . . . 5 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵))
7		an32 526	. . . . 5 ⊢ (((𝐹:𝐴–onto→𝑥 ∧ Fun ^◡𝐹) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
8	6, 7	bitri 182	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
9	8	exbii 1536	. . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
10		19.41v 1823	. . 3 ⊢ (∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
11	9, 10	bitri 182	. 2 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ^◡𝐹))
12	3, 4, 11	3bitr4i 210	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))

Colors of variables: wff set class

Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 ^◡ccnv 4362 Fun wfun 4916 ⟶wf 4918 –1-1→wf1 4919 –onto→wfo 4920 –1-1-onto→wf1o 4921

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-cnv 4371 df-dm 4373 df-rn 4374 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929

This theorem is referenced by: domen 6255

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