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| 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 6667 | . . 3 ⊢ (𝐹 Fn 𝐼 → Fun 𝐹) | |
| 2 | elunirn 7272 | . . 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 6670 | . . 3 ⊢ (𝐹 Fn 𝐼 → dom 𝐹 = 𝐼) | |
| 5 | 4 | rexeqdv 3326 | . 2 ⊢ (𝐹 Fn 𝐼 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) | 
| 6 | 3, 5 | bitrd 279 | 1 ⊢ (𝐹 Fn 𝐼 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 ∃wrex 3069 ∪ cuni 4906 dom cdm 5684 ran crn 5685 Fun wfun 6554 Fn wfn 6555 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: itunitc 10462 wunex2 10779 mreunirn 17645 arwhoma 18091 filunirn 23891 xmetunirn 24348 abfmpunirn 32663 cmpcref 33850 neibastop2lem 36362 stoweidlem59 46079 | 
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