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Mirrors > Home > ILE Home > Th. List > fssd | GIF version |
Description: Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fssd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
fssd.b | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
fssd | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fssd.b | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | fss 5220 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 406 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3021 ⟶wf 5055 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 df-f 5063 |
This theorem is referenced by: mapss 6515 ac6sfi 6721 fseq1p1m1 9715 resqrexlemcvg 10631 resqrexlemsqa 10636 climcvg1nlem 10957 fsumcl2lem 11006 ennnfonelemh 11709 cnrest2 12186 cnptoprest2 12190 cncfss 12483 limccnpcntop 12520 isomninnlem 12809 trilpolemisumle 12815 |
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