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Mirrors > Home > MPE Home > Th. List > f11o | Structured version Visualization version GIF version |
Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.) |
Ref | Expression |
---|---|
f11o.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
f11o | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f11o.1 | . . . 4 ⊢ 𝐹 ∈ V | |
2 | 1 | ffoss 7641 | . . 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 6355 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | dff1o3 6616 | . . . . . 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 644 | . . . . 5 ⊢ (((𝐹:𝐴–onto→𝑥 ∧ Fun ◡𝐹) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) | |
8 | 6, 7 | bitri 277 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) |
9 | 8 | exbii 1844 | . . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) |
10 | 19.41v 1946 | . . 3 ⊢ (∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) | |
11 | 9, 10 | bitri 277 | . 2 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) |
12 | 3, 4, 11 | 3bitr4i 305 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ◡ccnv 5549 Fun wfun 6344 ⟶wf 6346 –1-1→wf1 6347 –onto→wfo 6348 –1-1-onto→wf1o 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-cnv 5558 df-dm 5560 df-rn 5561 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 |
This theorem is referenced by: domen 8516 uspgrsprfo 44016 |
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