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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 2803 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 2 | brelrng 4961 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹) | |
| 3 | 2 | 3exp 1226 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))) |
| 4 | 1, 3 | mpi 15 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)) |
| 5 | elssuni 3919 | . . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹) | |
| 6 | 4, 5 | syl6 33 | . . 3 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
| 7 | 6 | alrimiv 1920 | . 2 ⊢ (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
| 8 | fvss 5649 | . 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 1393 ∈ wcel 2200 Vcvv 2800 ⊆ wss 3198 ∪ cuni 3891 class class class wbr 4086 ran crn 4724 ‘cfv 5324 |
| 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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2802 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-cnv 4731 df-dm 4733 df-rn 4734 df-iota 5284 df-fv 5332 |
| This theorem is referenced by: fvexg 5654 ovssunirng 6048 strfvssn 13094 ptex 13337 prdsvallem 13345 prdsval 13346 xmetunirn 15072 mopnval 15156 |
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