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Mirrors > Home > ILE Home > Th. List > relfvssunirn | GIF version |
Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
relfvssunirn | ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relelrn 4770 | . . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹) | |
2 | 1 | ex 114 | . . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)) |
3 | elssuni 3759 | . . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹) | |
4 | 2, 3 | syl6 33 | . . 3 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
5 | 4 | alrimiv 1846 | . 2 ⊢ (Rel 𝐹 → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹)) |
6 | fvss 5428 | . 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
7 | 5, 6 | syl 14 | 1 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 ∈ wcel 1480 ⊆ wss 3066 ∪ cuni 3731 class class class wbr 3924 ran crn 4535 Rel wrel 4539 ‘cfv 5118 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-iota 5083 df-fv 5126 |
This theorem is referenced by: relrnfvex 5432 |
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