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Mirrors > Home > ILE Home > Th. List > funiunfvdm | 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 5784. (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 5784 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹) | |
2 | imadmrn 4998 | . . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
3 | fndm 5334 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | imaeq2d 4988 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
5 | 2, 4 | eqtr3id 2236 | . . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴)) |
6 | 5 | unieqd 3835 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ (𝐹 “ 𝐴)) |
7 | 1, 6 | eqtrd 2222 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∪ cuni 3824 ∪ ciun 3901 dom cdm 4644 ran crn 4645 “ cima 4647 Fn wfn 5230 ‘cfv 5235 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 |
This theorem is referenced by: funiunfvdmf 5786 eluniimadm 5787 |
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