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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 5343 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3111 ⟶wf 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 df-f 5186 |
This theorem is referenced by: mapss 6648 ac6sfi 6855 fseq1p1m1 10019 resqrexlemcvg 10947 resqrexlemsqa 10952 climcvg1nlem 11276 fsumcl2lem 11325 ennnfonelemh 12274 cnrest2 12777 cnptoprest2 12781 cncfss 13111 limccnpcntop 13185 dvcoapbr 13212 dvef 13229 isomninnlem 13743 trilpolemisumle 13751 iswomninnlem 13762 ismkvnnlem 13765 |
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