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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 2741 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
2 | fvssunirng 5511 | . . 3 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑊 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
4 | rnexg 4876 | . . 3 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
5 | uniexg 4424 | . . 3 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V) |
7 | ssexg 4128 | . 2 ⊢ (((𝐹‘𝐴) ⊆ ∪ ran 𝐹 ∧ ∪ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) | |
8 | 3, 6, 7 | syl2anr 288 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 ∪ cuni 3796 ran crn 4612 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-cnv 4619 df-dm 4621 df-rn 4622 df-iota 5160 df-fv 5206 |
This theorem is referenced by: fvex 5516 ovexg 5887 rdgivallem 6360 frecabex 6377 mapsnconst 6672 cc2lem 7228 addvalex 7806 uzennn 10392 absval 10965 climmpt 11263 strnfvnd 12436 ressid 12479 grpsubval 12749 iscnp4 13012 cnpnei 13013 |
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