Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iunexg | Structured version Visualization version GIF version |
Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵. (Contributed by NM, 23-Mar-2006.) |
Ref | Expression |
---|---|
iunexg | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun2g 4488 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
3 | abrexexg 7034 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | |
4 | uniexg 6853 | . . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
7 | 2, 6 | eqeltrd 2688 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∪ cuni 4372 ∪ 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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: abrexex2g 7036 opabex3d 7037 opabex3 7038 iunex 7039 xpexgALT 7052 mpt2exxg 7133 ixpexg 7818 ixpssmapg 7824 iundom 9243 iunctb 9275 cshwsex 15645 imasplusg 16000 imasmulr 16001 imasvsca 16003 imasip 16004 gsum2d2 18196 gsumcom2 18197 dprd2da 18264 ptcls 21229 ptcmplem2 21667 elpwiuncl 28743 aciunf1lem 28844 esum2dlem 29481 esum2d 29482 esumiun 29483 omssubadd 29689 eulerpartlemgs2 29769 bnj535 30214 bnj546 30220 bnj893 30252 bnj1136 30319 bnj1413 30357 trpredtr 30974 trpredmintr 30975 trpredrec 30982 eliunov2 36990 fvmptiunrelexplb0d 36995 fvmptiunrelexplb1d 36997 iunrelexp0 37013 unirnmapsn 38401 iunmapss 38402 ssmapsn 38403 iunmapsn 38404 sge0iunmptlemfi 39306 sge0iunmpt 39311 smflimlem1 39657 mpt2exxg2 41909 |
Copyright terms: Public domain | W3C validator |