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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 7879 | . . 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 6502 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | dff1o3 6791 | . . . . . 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 645 | . . . . 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 1851 | . . 3 ⊢ (∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ∃𝑥((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ Fun ◡𝐹)) |
10 | 19.41v 1954 | . . 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 205 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 ◡ccnv 5633 Fun wfun 6491 ⟶wf 6493 –1-1→wf1 6494 –onto→wfo 6495 –1-1-onto→wf1o 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-cnv 5642 df-dm 5644 df-rn 5645 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: domen 8904 uspgrsprfo 46136 |
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