![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33397 | . . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | 1 | tfr2 8017 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴))) |
3 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) | |
4 | rneq 5770 | . . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴)) | |
5 | df-ima 5532 | . . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴) | |
6 | 4, 5 | eqtr4di 2851 | . . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴)) |
7 | 6 | unieqd 4814 | . . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴)) |
8 | 7 | pweqd 4516 | . . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴)) |
9 | 8 | sqxpeqd 5551 | . . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) |
10 | 9 | imaeq2d 5896 | . . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
11 | 1 | tfr1 8016 | . . . . 5 ⊢ M Fn On |
12 | fnfun 6423 | . . . . 5 ⊢ ( M Fn On → Fun M ) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun M |
14 | resfunexg 6955 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V) | |
15 | 13, 14 | mpan 689 | . . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V) |
16 | scutf 33386 | . . . . 5 ⊢ |s : <<s ⟶ No | |
17 | ffun 6490 | . . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s ) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ Fun |s |
19 | funimaexg 6410 | . . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
20 | 13, 19 | mpan 689 | . . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
21 | uniexg 7446 | . . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V) | |
22 | pwexg 5244 | . . . . . 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 7454 | . . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) |
25 | funimaexg 6410 | . . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) | |
26 | 18, 24, 25 | sylancr 590 | . . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) |
27 | 3, 10, 15, 26 | fvmptd3 6768 | . 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 2833 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 ∪ cuni 4800 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ↾ cres 5521 “ cima 5522 Oncon0 6159 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 No csur 33260 <<s csslt 33363 |s cscut 33365 M cmade 33392 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-wrecs 7930 df-recs 7991 df-1o 8085 df-2o 8086 df-no 33263 df-slt 33264 df-bday 33265 df-sslt 33364 df-scut 33366 df-made 33397 |
This theorem is referenced by: madeval2 33403 |
Copyright terms: Public domain | W3C validator |