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Mirrors > Home > ILE Home > Th. List > fvexg | GIF version |
Description: Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.) |
Ref | Expression |
---|---|
fvexg | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
2 | fvssunirng 5549 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑊 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
4 | rnexg 4910 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
5 | uniexg 4457 | . . 3 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) |
7 | ssexg 4157 | . 2 ⊢ (((𝐹‘𝐴) ⊆ ∪ ran 𝐹 ∧ ∪ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) | |
8 | 3, 6, 7 | syl2anr 290 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ∪ cuni 3824 ran crn 4645 ‘cfv 5235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-cnv 4652 df-dm 4654 df-rn 4655 df-iota 5196 df-fv 5243 |
This theorem is referenced by: fvex 5554 ovexg 5930 rdgivallem 6406 frecabex 6423 mapsnconst 6720 cc2lem 7295 addvalex 7873 uzennn 10467 absval 11042 climmpt 11340 strnfvnd 12532 prdsex 12774 imasex 12782 imasival 12783 imasbas 12784 imasplusg 12785 imasmulr 12786 imasaddfnlemg 12791 imasaddvallemg 12792 grpsubval 12990 mulgval 13064 mulgfng 13066 znval 13932 znle 13933 znbaslemnn 13935 znbas 13939 iscnp4 14175 cnpnei 14176 |
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