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| Mirrors > Home > MPE Home > Th. List > iunex | Structured version Visualization version GIF version | ||
| Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be read informally as 𝐵(𝑥). (Contributed by NM, 13-Oct-2003.) |
| Ref | Expression |
|---|---|
| iunex.1 | ⊢ 𝐴 ∈ V |
| iunex.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| iunex | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunex.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | iunex.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 2 | rgenw 3065 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
| 4 | iunexg 7988 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-uni 4908 df-iun 4993 |
| This theorem is referenced by: tz9.1 9769 tz9.1c 9770 cplem2 9930 fseqdom 10066 pwsdompw 10243 cfsmolem 10310 ac6c4 10521 konigthlem 10608 alephreg 10622 pwfseqlem4 10702 pwfseqlem5 10703 pwxpndom2 10705 wunex2 10778 wuncval2 10787 inar1 10815 rtrclreclem1 15096 dfrtrclrec2 15097 rtrclreclem2 15098 rtrclreclem4 15100 isfunc 17909 smndex1bas 18919 smndex1sgrp 18921 smndex1mnd 18923 smndex1id 18924 dfac14 23626 txcmplem2 23650 cnextfval 24070 bnj893 34942 colinearex 36061 volsupnfl 37672 heiborlem3 37820 comptiunov2i 43719 corclrcl 43720 iunrelexpmin1 43721 trclrelexplem 43724 iunrelexpmin2 43725 dftrcl3 43733 trclfvcom 43736 cnvtrclfv 43737 cotrcltrcl 43738 trclimalb2 43739 trclfvdecomr 43741 dfrtrcl3 43746 dfrtrcl4 43751 corcltrcl 43752 cotrclrcl 43755 carageniuncllem1 46536 carageniuncllem2 46537 carageniuncl 46538 caratheodorylem1 46541 caratheodorylem2 46542 ovnovollem1 46671 ovnovollem2 46672 smfresal 46803 |
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