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Mirrors > Home > ILE Home > Th. List > funiunfvdm | Structured version GIF version |
Description: The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5324. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Ref | Expression |
---|---|
funiunfvdm | ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fniunfv 5324 | . 2 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹) | |
2 | imadmrn 4603 | . . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
3 | fndm 4922 | . . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A) | |
4 | 3 | imaeq2d 4593 | . . . 4 ⊢ (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹 “ A)) |
5 | 2, 4 | syl5eqr 2069 | . . 3 ⊢ (𝐹 Fn A → ran 𝐹 = (𝐹 “ A)) |
6 | 5 | unieqd 3564 | . 2 ⊢ (𝐹 Fn A → ∪ ran 𝐹 = ∪ (𝐹 “ A)) |
7 | 1, 6 | eqtrd 2055 | 1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1228 ∪ cuni 3553 ∪ ciun 3630 dom cdm 4270 ran crn 4271 “ cima 4273 Fn wfn 4822 ‘cfv 4827 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1364 ax-ie2 1365 ax-8 1377 ax-10 1378 ax-11 1379 ax-i12 1380 ax-bnd 1381 ax-4 1382 ax-14 1387 ax-17 1401 ax-i9 1405 ax-ial 1410 ax-i5r 1411 ax-ext 2005 ax-sep 3848 ax-pow 3900 ax-pr 3917 |
This theorem depends on definitions: df-bi 110 df-3an 875 df-tru 1231 df-nf 1330 df-sb 1629 df-eu 1886 df-mo 1887 df-clab 2010 df-cleq 2016 df-clel 2019 df-nfc 2150 df-ral 2288 df-rex 2289 df-v 2536 df-sbc 2741 df-un 2898 df-in 2900 df-ss 2907 df-pw 3335 df-sn 3355 df-pr 3356 df-op 3358 df-uni 3554 df-iun 3632 df-br 3738 df-opab 3792 df-mpt 3793 df-id 4003 df-xp 4276 df-rel 4277 df-cnv 4278 df-co 4279 df-dm 4280 df-rn 4281 df-res 4282 df-ima 4283 df-iota 4792 df-fun 4829 df-fn 4830 df-fv 4835 |
This theorem is referenced by: funiunfvdmf 5326 eluniimadm 5327 |
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