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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 5372 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) | |
2 | 3anass 966 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))) | |
3 | df-rn 4550 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
4 | 3 | eqeq1i 2147 | . . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ◡𝐹 = 𝐵) |
5 | 4 | anbi2i 452 | . . . 4 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵)) |
6 | df-fn 5126 | . . . 4 ⊢ (◡𝐹 Fn 𝐵 ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵)) | |
7 | 5, 6 | bitr4i 186 | . . 3 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ◡𝐹 Fn 𝐵) |
8 | 7 | anbi2i 452 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
9 | 1, 2, 8 | 3bitri 205 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ◡ccnv 4538 dom cdm 4539 ran crn 4540 Fun wfun 5117 Fn wfn 5118 –1-1-onto→wf1o 5122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 df-rn 4550 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 |
This theorem is referenced by: f1ocnv 5380 f1oun 5387 f1o00 5402 f1oi 5405 f1osn 5407 f1ompt 5571 f1ofveu 5762 f1ocnvd 5972 f1od2 6132 mapsnf1o2 6590 sbthlemi9 6853 hmeof1o2 12477 |
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