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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 5715 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 2299 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
3 | 1, 2 | nffv 5480 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
4 | nfcv 2299 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) | |
5 | fveq2 5470 | . . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
6 | 3, 4, 5 | cbviun 3888 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) |
7 | funiunfvdm 5715 | . 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑧 ∈ 𝐴 (𝐹‘𝑧) = ∪ (𝐹 “ 𝐴)) | |
8 | 6, 7 | eqtr3id 2204 | 1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 Ⅎwnfc 2286 ∪ cuni 3774 ∪ ciun 3851 “ cima 4591 Fn wfn 5167 ‘cfv 5172 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-fv 5180 |
This theorem is referenced by: (None) |
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