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Mirrors > Home > ILE Home > Th. List > dff1o4 | GIF version |
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
dff1o4 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o2 5485 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) | |
2 | 3anass 984 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))) | |
3 | df-rn 4655 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
4 | 3 | eqeq1i 2197 | . . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ◡𝐹 = 𝐵) |
5 | 4 | anbi2i 457 | . . . 4 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵)) |
6 | df-fn 5238 | . . . 4 ⊢ (◡𝐹 Fn 𝐵 ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵)) | |
7 | 5, 6 | bitr4i 187 | . . 3 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ◡𝐹 Fn 𝐵) |
8 | 7 | anbi2i 457 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
9 | 1, 2, 8 | 3bitri 206 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ◡ccnv 4643 dom cdm 4644 ran crn 4645 Fun wfun 5229 Fn wfn 5230 –1-1-onto→wf1o 5234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 df-rn 4655 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 |
This theorem is referenced by: f1ocnv 5493 f1oun 5500 f1o00 5515 f1oi 5518 f1osn 5520 f1ompt 5688 f1ofveu 5885 f1ocnvd 6097 f1od2 6261 mapsnf1o2 6723 sbthlemi9 6995 mhmf1o 12937 grpinvf1o 13029 ghmf1o 13231 rhmf1o 13535 hmeof1o2 14285 |
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