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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 3150 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
4 | iunexg 7658 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
5 | 1, 3, 4 | mp2an 690 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∪ ciun 4911 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: tz9.1 9165 tz9.1c 9166 cplem2 9313 fseqdom 9446 pwsdompw 9620 cfsmolem 9686 ac6c4 9897 konigthlem 9984 alephreg 9998 pwfseqlem4 10078 pwfseqlem5 10079 pwxpndom2 10081 wunex2 10154 wuncval2 10163 inar1 10191 dfrtrclrec2 14410 rtrclreclem1 14411 rtrclreclem2 14412 rtrclreclem4 14414 isfunc 17128 smndex1bas 18065 smndex1sgrp 18067 smndex1mnd 18069 smndex1id 18070 dfac14 22220 txcmplem2 22244 cnextfval 22664 bnj893 32195 colinearex 33516 volsupnfl 34931 heiborlem3 35085 comptiunov2i 40044 corclrcl 40045 iunrelexpmin1 40046 trclrelexplem 40049 iunrelexpmin2 40050 dftrcl3 40058 trclfvcom 40061 cnvtrclfv 40062 cotrcltrcl 40063 trclimalb2 40064 trclfvdecomr 40066 dfrtrcl3 40071 dfrtrcl4 40076 corcltrcl 40077 cotrclrcl 40080 carageniuncllem1 42797 carageniuncllem2 42798 carageniuncl 42799 caratheodorylem1 42802 caratheodorylem2 42803 ovnovollem1 42932 ovnovollem2 42933 smfresal 43057 |
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