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Mirrors > Home > ILE Home > Th. List > imaexg | GIF version |
Description: The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.) |
Ref | Expression |
---|---|
imaexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 4743 | . 2 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 | |
2 | rnexg 4659 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | |
3 | ssexg 3946 | . 2 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V) | |
4 | 1, 2, 3 | sylancr 405 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1436 Vcvv 2614 ⊆ wss 2986 ran crn 4405 “ cima 4407 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-v 2616 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-xp 4410 df-cnv 4412 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 |
This theorem is referenced by: imaex 4745 ecexg 6229 fopwdom 6485 isinfinf 6546 |
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