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Mirrors > Home > MPE Home > Th. List > itunifval | Structured version Visualization version GIF version |
Description: Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could conceivably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
ituni.u | ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω)) |
Ref | Expression |
---|---|
itunifval | ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3492 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | rdgeq2 8411 | . . . 4 ⊢ (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴)) | |
3 | 2 | reseq1d 5980 | . . 3 ⊢ (𝑥 = 𝐴 → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)) |
4 | ituni.u | . . 3 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω)) | |
5 | rdgfun 8415 | . . . 4 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) | |
6 | omex 9637 | . . . 4 ⊢ ω ∈ V | |
7 | resfunexg 7216 | . . . 4 ⊢ ((Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V) | |
8 | 5, 6, 7 | mp2an 690 | . . 3 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V |
9 | 3, 4, 8 | fvmpt 6998 | . 2 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)) |
10 | 1, 9 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∪ cuni 4908 ↦ cmpt 5231 ↾ cres 5678 Fun wfun 6537 ‘cfv 6543 ωcom 7854 reccrdg 8408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 |
This theorem is referenced by: itunifn 10411 ituni0 10412 itunisuc 10413 |
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