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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 7969 | . . 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 6568 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | dff1o3 6855 | . . . . . 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 1845 | . . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) |
10 | 19.41v 1947 | . . 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 1776 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ◡ccnv 5688 Fun wfun 6557 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: domen 9001 uspgrsprfo 47992 |
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