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Mirrors > Home > ILE Home > Th. List > funiunfvdmf | GIF version |
Description: The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5664 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
Ref | Expression |
---|---|
funiunfvf.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
funiunfvdmf | ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funiunfvf.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
3 | 1, 2 | nffv 5431 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
4 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
5 | fveq2 5421 | . . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
6 | 3, 4, 5 | cbviun 3850 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) |
7 | funiunfvdm 5664 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴)) | |
8 | 6, 7 | syl5eqr 2186 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Ⅎwnfc 2268 ∪ cuni 3736 ∪ ciun 3813 “ cima 4542 Fn wfn 5118 ‘cfv 5123 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: (None) |
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