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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 2953 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
4 | iunexg 7185 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
5 | 1, 3, 4 | mp2an 708 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ∪ ciun 4552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 |
This theorem is referenced by: abrexex2OLD 7192 tz9.1 8643 tz9.1c 8644 cplem2 8791 fseqdom 8887 pwsdompw 9064 cfsmolem 9130 ac6c4 9341 konigthlem 9428 alephreg 9442 pwfseqlem4 9522 pwfseqlem5 9523 pwxpndom2 9525 wunex2 9598 wuncval2 9607 inar1 9635 dfrtrclrec2 13841 rtrclreclem1 13842 rtrclreclem2 13843 rtrclreclem4 13845 isfunc 16571 dfac14 21469 txcmplem2 21493 cnextfval 21913 bnj893 31124 colinearex 32292 volsupnfl 33584 heiborlem3 33742 comptiunov2i 38315 corclrcl 38316 iunrelexpmin1 38317 trclrelexplem 38320 iunrelexpmin2 38321 dftrcl3 38329 trclfvcom 38332 cnvtrclfv 38333 cotrcltrcl 38334 trclimalb2 38335 trclfvdecomr 38337 dfrtrcl3 38342 dfrtrcl4 38347 corcltrcl 38348 cotrclrcl 38351 carageniuncllem1 41056 carageniuncllem2 41057 carageniuncl 41058 caratheodorylem1 41061 caratheodorylem2 41062 ovnovollem1 41191 ovnovollem2 41192 smfresal 41316 |
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