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Mirrors > Home > ILE Home > Th. List > fvss | GIF version |
Description: The value of a function is a subset of 𝐵 if every element that could be a candidate for the value is a subset of 𝐵. (Contributed by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
fvss | ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 5139 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
2 | iotass 5113 | . 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (℩𝑥𝐴𝐹𝑥) ⊆ 𝐵) | |
3 | 1, 2 | eqsstrid 3148 | 1 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 ⊆ wss 3076 class class class wbr 3937 ℩cio 5094 ‘cfv 5131 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-iota 5096 df-fv 5139 |
This theorem is referenced by: fvssunirng 5444 relfvssunirn 5445 sefvex 5450 fvmptss2 5504 tfrexlem 6239 |
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