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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 3465 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | rdgeq2 8362 | . . . 4 ⊢ (𝑥 = 𝐴 → rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) = rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴)) | |
3 | 2 | reseq1d 5940 | . . 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 8366 | . . . 4 ⊢ Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) | |
6 | omex 9587 | . . . 4 ⊢ ω ∈ V | |
7 | resfunexg 7169 | . . . 4 ⊢ ((Fun rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ∧ ω ∈ V) → (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V) | |
8 | 5, 6, 7 | mp2an 691 | . . 3 ⊢ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω) ∈ V |
9 | 3, 4, 8 | fvmpt 6952 | . 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 1542 ∈ wcel 2107 Vcvv 3447 ∪ cuni 4869 ↦ cmpt 5192 ↾ cres 5639 Fun wfun 6494 ‘cfv 6500 ωcom 7806 reccrdg 8359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 |
This theorem is referenced by: itunifn 10361 ituni0 10362 itunisuc 10363 |
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