![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f1ss | Structured version Visualization version GIF version |
Description: A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1ss | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6242 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | fss 6197 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | sylan 563 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
4 | df-f1 6037 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | 4 | simprbi 480 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
6 | 5 | adantr 466 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → Fun ◡𝐹) |
7 | df-f1 6037 | . 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹)) | |
8 | 3, 6, 7 | sylanbrc 566 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ⊆ wss 3724 ◡ccnv 5249 Fun wfun 6026 ⟶wf 6028 –1-1→wf1 6029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-in 3731 df-ss 3738 df-f 6036 df-f1 6037 |
This theorem is referenced by: f1sng 6320 f1prex 6683 domssex2 8277 1sdom 8320 marypha1lem 8496 marypha2 8502 isinffi 9019 fseqenlem1 9048 dfac12r 9171 ackbij2 9268 cff1 9283 fin23lem28 9365 fin23lem41 9377 pwfseqlem5 9688 hashf1lem1 13442 gsumzres 18518 gsumzcl2 18519 gsumzf1o 18521 gsumzaddlem 18529 gsumzmhm 18545 gsumzoppg 18552 lindfres 20380 islindf3 20383 dvne0f1 23996 istrkg2ld 25581 ausgrusgrb 26283 uspgrushgr 26293 usgruspgr 26296 uspgr1e 26360 sizusglecusglem1 26593 qqhre 30405 erdsze2lem1 31524 eldioph2lem2 37851 eldioph2 37852 |
Copyright terms: Public domain | W3C validator |