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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 6568 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | fss 6520 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) |
4 | df-f1 6353 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
5 | 4 | simprbi 497 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → Fun ◡𝐹) |
7 | df-f1 6353 | . 2 ⊢ (𝐹:𝐴–1-1→𝐶 ↔ (𝐹:𝐴⟶𝐶 ∧ Fun ◡𝐹)) | |
8 | 3, 6, 7 | sylanbrc 583 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ⊆ wss 3933 ◡ccnv 5547 Fun wfun 6342 ⟶wf 6344 –1-1→wf1 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-f 6352 df-f1 6353 |
This theorem is referenced by: f1sng 6649 f1prex 7031 domssex2 8665 1sdom 8709 marypha1lem 8885 marypha2 8891 isinffi 9409 fseqenlem1 9438 dfac12r 9560 ackbij2 9653 cff1 9668 fin23lem28 9750 fin23lem41 9762 pwfseqlem5 10073 hashf1lem1 13801 gsumzres 18958 gsumzcl2 18959 gsumzf1o 18961 gsumzaddlem 18970 gsumzmhm 18986 gsumzoppg 18993 lindfres 20895 islindf3 20898 dvne0f1 24536 istrkg2ld 26173 ausgrusgrb 26877 uspgrushgr 26887 usgruspgr 26890 uspgr1e 26953 sizusglecusglem1 27170 s1f1 30546 s2f1 30548 qqhre 31160 erdsze2lem1 32347 eldioph2lem2 39236 eldioph2 39237 fundcmpsurbijinjpreimafv 43444 fundcmpsurinjimaid 43448 |
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