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Mirrors > Home > MPE Home > Th. List > iuneq2i | Structured version Visualization version GIF version |
Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
iuneq2i.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iuneq2i | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq2 4473 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | |
2 | iuneq2i.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
3 | 1, 2 | mprg 2910 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-iun 4457 |
This theorem is referenced by: dfiunv2 4492 iunrab 4503 iunid 4511 iunin1 4521 2iunin 4524 resiun1 5336 resiun1OLD 5337 resiun2 5338 dfimafn2 6156 dfmpt 6316 funiunfv 6410 fpar 7168 onovuni 7326 uniqs 7694 marypha2lem2 8225 alephlim 8773 cfsmolem 8975 ituniiun 9127 imasdsval2 15999 lpival 19066 cmpsublem 21012 txbasval 21219 uniioombllem2 23157 uniioombllem4 23160 volsup2 23179 itg1addlem5 23273 itg1climres 23287 indval2 29404 sigaclfu2 29511 measvuni 29604 trpred0 30980 rabiun 32552 mblfinlem2 32617 voliunnfl 32623 cnambfre 32628 trclrelexplem 37022 cotrclrcl 37053 hoicvr 39438 hoidmv1le 39484 hoidmvle 39490 hspmbllem2 39517 smflimlem3 39659 smflimlem4 39660 smflim 39663 dfaimafn2 39895 xpiun 41556 |
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