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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 27180 | . . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | 1 | tfr2 8345 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴))) |
3 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) | |
4 | rneq 5892 | . . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴)) | |
5 | df-ima 5647 | . . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴) | |
6 | 4, 5 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴)) |
7 | 6 | unieqd 4880 | . . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴)) |
8 | 7 | pweqd 4578 | . . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴)) |
9 | 8 | sqxpeqd 5666 | . . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) |
10 | 9 | imaeq2d 6014 | . . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
11 | 1 | tfr1 8344 | . . . . 5 ⊢ M Fn On |
12 | fnfun 6603 | . . . . 5 ⊢ ( M Fn On → Fun M ) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun M |
14 | resfunexg 7166 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V) | |
15 | 13, 14 | mpan 689 | . . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V) |
16 | scutf 27154 | . . . . 5 ⊢ |s : <<s ⟶ No | |
17 | ffun 6672 | . . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s ) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ Fun |s |
19 | funimaexg 6588 | . . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
20 | 13, 19 | mpan 689 | . . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
21 | uniexg 7678 | . . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V) | |
22 | pwexg 5334 | . . . . . 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 7686 | . . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) |
25 | funimaexg 6588 | . . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) | |
26 | 18, 24, 25 | sylancr 588 | . . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) |
27 | 3, 10, 15, 26 | fvmptd3 6972 | . 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 2777 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3446 𝒫 cpw 4561 ∪ cuni 4866 ↦ cmpt 5189 × cxp 5632 ran crn 5635 ↾ cres 5636 “ cima 5637 Oncon0 6318 Fun wfun 6491 Fn wfn 6492 ⟶wf 6493 ‘cfv 6497 No csur 26991 <<s csslt 27123 |s cscut 27125 M cmade 27175 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-made 27180 |
This theorem is referenced by: madeval2 27186 |
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