| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > newval | Structured version Visualization version GIF version | ||
| Description: The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.) |
| Ref | Expression |
|---|---|
| newval | ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴)) | |
| 2 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴)) | |
| 3 | 1, 2 | difeq12d 4127 | . . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
| 4 | df-new 27888 | . . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥))) | |
| 5 | fvex 6919 | . . . 4 ⊢ ( M ‘𝐴) ∈ V | |
| 6 | 5 | difexi 5330 | . . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V |
| 7 | 3, 4, 6 | fvmpt 7016 | . 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
| 8 | 4 | fvmptndm 7047 | . . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅) |
| 9 | df-made 27886 | . . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓)))) | |
| 10 | 9 | tfr1 8437 | . . . . . . . 8 ⊢ M Fn On |
| 11 | 10 | fndmi 6672 | . . . . . . 7 ⊢ dom M = On |
| 12 | 11 | eleq2i 2833 | . . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On) |
| 13 | ndmfv 6941 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅) | |
| 14 | 12, 13 | sylnbir 331 | . . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅) |
| 15 | 14 | difeq1d 4125 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴))) |
| 16 | 0dif 4405 | . . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅ | |
| 17 | 15, 16 | eqtrdi 2793 | . . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅) |
| 18 | 8, 17 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
| 19 | 7, 18 | pm2.61i 182 | 1 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ran crn 5686 “ cima 5688 Oncon0 6384 ‘cfv 6561 |s cscut 27827 M cmade 27881 O cold 27882 N cnew 27883 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-made 27886 df-new 27888 |
| This theorem is referenced by: new0 27913 madeun 27922 newbday 27940 |
| Copyright terms: Public domain | W3C validator |