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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 33284 | . . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | 1 | tfr2 8034 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))‘( M ↾ 𝐴))) |
3 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) = (𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))) | |
4 | rneq 5806 | . . . . . . . 8 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ran ( M ↾ 𝐴)) | |
5 | df-ima 5568 | . . . . . . . 8 ⊢ ( M “ 𝐴) = ran ( M ↾ 𝐴) | |
6 | 4, 5 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑥 = ( M ↾ 𝐴) → ran 𝑥 = ( M “ 𝐴)) |
7 | 6 | unieqd 4852 | . . . . . 6 ⊢ (𝑥 = ( M ↾ 𝐴) → ∪ ran 𝑥 = ∪ ( M “ 𝐴)) |
8 | 7 | pweqd 4558 | . . . . 5 ⊢ (𝑥 = ( M ↾ 𝐴) → 𝒫 ∪ ran 𝑥 = 𝒫 ∪ ( M “ 𝐴)) |
9 | 8 | sqxpeqd 5587 | . . . 4 ⊢ (𝑥 = ( M ↾ 𝐴) → (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥) = (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) |
10 | 9 | imaeq2d 5929 | . . 3 ⊢ (𝑥 = ( M ↾ 𝐴) → ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
11 | 1 | tfr1 8033 | . . . . 5 ⊢ M Fn On |
12 | fnfun 6453 | . . . . 5 ⊢ ( M Fn On → Fun M ) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun M |
14 | resfunexg 6978 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M ↾ 𝐴) ∈ V) | |
15 | 13, 14 | mpan 688 | . . 3 ⊢ (𝐴 ∈ On → ( M ↾ 𝐴) ∈ V) |
16 | scutf 33273 | . . . . 5 ⊢ |s : <<s ⟶ No | |
17 | ffun 6517 | . . . . 5 ⊢ ( |s : <<s ⟶ No → Fun |s ) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ Fun |s |
19 | funimaexg 6440 | . . . . . . 7 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
20 | 13, 19 | mpan 688 | . . . . . 6 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
21 | uniexg 7466 | . . . . . 6 ⊢ (( M “ 𝐴) ∈ V → ∪ ( M “ 𝐴) ∈ V) | |
22 | pwexg 5279 | . . . . . 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 7474 | . . . 4 ⊢ (𝐴 ∈ On → (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) |
25 | funimaexg 6440 | . . . 4 ⊢ ((Fun |s ∧ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)) ∈ V) → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) | |
26 | 18, 24, 25 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ On → ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴))) ∈ V) |
27 | 3, 10, 15, 26 | fvmptd3 6791 | . 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 2856 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 𝒫 cpw 4539 ∪ cuni 4838 ↦ cmpt 5146 × cxp 5553 ran crn 5556 ↾ cres 5557 “ cima 5558 Oncon0 6191 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 No csur 33147 <<s csslt 33250 |s cscut 33252 M cmade 33279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7947 df-recs 8008 df-1o 8102 df-2o 8103 df-no 33150 df-slt 33151 df-bday 33152 df-sslt 33251 df-scut 33253 df-made 33284 |
This theorem is referenced by: madeval2 33290 |
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