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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 5284 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3071 ⟶wf 5119 |
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-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 df-f 5127 |
This theorem is referenced by: mapss 6585 ac6sfi 6792 fseq1p1m1 9874 resqrexlemcvg 10791 resqrexlemsqa 10796 climcvg1nlem 11118 fsumcl2lem 11167 ennnfonelemh 11917 cnrest2 12405 cnptoprest2 12409 cncfss 12739 limccnpcntop 12813 dvcoapbr 12840 dvef 12856 isomninnlem 13225 trilpolemisumle 13231 |
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