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Mirrors > Home > MPE Home > Th. List > fnunirn | Structured version Visualization version GIF version |
Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
fnunirn | ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6432 | . . 3 ⊢ (𝐹 Fn 𝐼 → Fun 𝐹) | |
2 | elunirn 7015 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
4 | fndm 6434 | . . 3 ⊢ (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼) | |
5 | 4 | rexeqdv 3316 | . 2 ⊢ (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) |
6 | 3, 5 | bitrd 282 | 1 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2113 ∃wrex 3054 ∪ cuni 4793 dom cdm 5519 ran crn 5520 Fun wfun 6327 Fn wfn 6328 ‘cfv 6333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-fv 6341 |
This theorem is referenced by: itunitc 9914 wunex2 10231 mreunirn 16968 arwhoma 17410 filunirn 22626 xmetunirn 23083 abfmpunirn 30556 cmpcref 31364 neibastop2lem 34179 stoweidlem59 43126 |
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