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Mirrors > Home > MPE Home > Th. List > madeval | Structured version Visualization version GIF version |
Description: The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.) |
Ref | Expression |
---|---|
madeval | ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-made 27820 | . . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | 1 | tfr2 8419 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴))) |
3 | eqid 2725 | . . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) | |
4 | rneq 5938 | . . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴)) | |
5 | df-ima 5691 | . . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴) | |
6 | 4, 5 | eqtr4di 2783 | . . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴)) |
7 | 6 | unieqd 4922 | . . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴)) |
8 | 7 | pweqd 4621 | . . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴)) |
9 | 8 | sqxpeqd 5710 | . . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) |
10 | 9 | imaeq2d 6064 | . . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
11 | 1 | tfr1 8418 | . . . . 5 ⊢ M Fn On |
12 | fnfun 6655 | . . . . 5 ⊢ ( M Fn On → Fun M ) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun M |
14 | resfunexg 7227 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V) | |
15 | 13, 14 | mpan 688 | . . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V) |
16 | scutf 27791 | . . . . 5 ⊢ |s : <<s ⟶ No | |
17 | ffun 6726 | . . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s ) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ Fun |s |
19 | funimaexg 6640 | . . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
20 | 13, 19 | mpan 688 | . . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
21 | uniexg 7746 | . . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V) | |
22 | pwexg 5378 | . . . . . 6 ⊢ (∪ ( M “ 𝐴) ∈ V → 𝒫 ∪ ( M “ 𝐴) ∈ V) | |
23 | 20, 21, 22 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ On → 𝒫 ∪ ( M “ 𝐴) ∈ V) |
24 | 23, 23 | xpexd 7754 | . . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) |
25 | funimaexg 6640 | . . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) | |
26 | 18, 24, 25 | sylancr 585 | . . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) |
27 | 3, 10, 15, 26 | fvmptd3 7027 | . 2 ⊢ (𝐴 ∈ On → ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
28 | 2, 27 | eqtrd 2765 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 𝒫 cpw 4604 ∪ cuni 4909 ↦ cmpt 5232 × cxp 5676 ran crn 5679 ↾ cres 5680 “ cima 5681 Oncon0 6371 Fun wfun 6543 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 No csur 27618 <<s csslt 27759 |s cscut 27761 M cmade 27815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-1o 8487 df-2o 8488 df-no 27621 df-slt 27622 df-bday 27623 df-sslt 27760 df-scut 27762 df-made 27820 |
This theorem is referenced by: madeval2 27826 |
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