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Mirrors > Home > ILE Home > Th. List > fvssunirng | GIF version |
Description: The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
fvssunirng | ⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | brelrng 4778 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹) | |
3 | 2 | 3exp 1181 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))) |
4 | 1, 3 | mpi 15 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)) |
5 | elssuni 3772 | . . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹) | |
6 | 4, 5 | syl6 33 | . . 3 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
7 | 6 | alrimiv 1847 | . 2 ⊢ (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
8 | fvss 5443 | . 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
9 | 7, 8 | syl 14 | 1 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 ∪ cuni 3744 class class class wbr 3937 ran crn 4548 ‘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-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 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-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-cnv 4555 df-dm 4557 df-rn 4558 df-iota 5096 df-fv 5139 |
This theorem is referenced by: fvexg 5448 strfvssn 12020 xmetunirn 12566 mopnval 12650 |
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