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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 4954 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
3 | abrexexg 7661 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) | |
4 | 3 | uniexd 7467 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
6 | 2, 5 | eqeltrd 2913 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cuni 4837 ∪ ciun 4918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 |
This theorem is referenced by: abrexex2g 7664 opabex3d 7665 opabex3rd 7666 opabex3 7667 iunex 7668 xpexgALT 7681 mpoexxg 7772 ixpexg 8485 ixpssmapg 8491 iundom 9963 iunctb 9995 wrdexg 13870 cshwsex 16433 imasplusg 16789 imasmulr 16790 imasvsca 16792 imasip 16793 gsum2d2 19093 gsumcom2 19094 dprd2da 19163 ptcls 22223 ptcmplem2 22660 elpwiuncl 30287 aciunf1lem 30406 esum2dlem 31351 esum2d 31352 esumiun 31353 omssubadd 31558 eulerpartlemgs2 31638 bnj535 32162 bnj546 32168 bnj893 32200 bnj1136 32269 bnj1413 32307 trpredtr 33069 trpredmintr 33070 trpredrec 33077 eliunov2 40022 fvmptiunrelexplb0d 40027 fvmptiunrelexplb1d 40029 iunrelexp0 40045 collexd 40591 unirnmapsn 41475 iunmapss 41476 ssmapsn 41477 iunmapsn 41478 sge0iunmptlemfi 42694 sge0iunmpt 42699 smflimlem1 43046 smfliminflem 43103 mpoexxg2 44385 |
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