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| Mirrors > Home > MPE Home > Th. List > madeoldsuc | Structured version Visualization version GIF version | ||
| Description: The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.) |
| Ref | Expression |
|---|---|
| madeoldsuc | ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-suc 6390 | . . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
| 2 | 1 | imaeq2i 6076 | . . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴})) |
| 3 | imaundi 6169 | . . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴})) | |
| 4 | 2, 3 | eqtri 2765 | . . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴})) |
| 5 | 4 | unieqi 4919 | . . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) |
| 6 | uniun 4930 | . . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) | |
| 7 | 5, 6 | eqtri 2765 | . . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) |
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))) |
| 9 | oldval 27893 | . . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) | |
| 10 | 9 | eqcomd 2743 | . . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴)) |
| 11 | madef 27895 | . . . . . . . 8 ⊢ M :On⟶𝒫 No | |
| 12 | ffn 6736 | . . . . . . . 8 ⊢ ( M :On⟶𝒫 No → M Fn On) | |
| 13 | 11, 12 | ax-mp 5 | . . . . . . 7 ⊢ M Fn On |
| 14 | fnsnfv 6988 | . . . . . . . 8 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → {( M ‘𝐴)} = ( M “ {𝐴})) | |
| 15 | 14 | eqcomd 2743 | . . . . . . 7 ⊢ (( M Fn On ∧ 𝐴 ∈ On) → ( M “ {𝐴}) = {( M ‘𝐴)}) |
| 16 | 13, 15 | mpan 690 | . . . . . 6 ⊢ (𝐴 ∈ On → ( M “ {𝐴}) = {( M ‘𝐴)}) |
| 17 | 16 | unieqd 4920 | . . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)}) |
| 18 | fvex 6919 | . . . . . 6 ⊢ ( M ‘𝐴) ∈ V | |
| 19 | 18 | unisn 4926 | . . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴) |
| 20 | 17, 19 | eqtrdi 2793 | . . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴)) |
| 21 | 10, 20 | uneq12d 4169 | . . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴))) |
| 22 | oldssmade 27916 | . . . . 5 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) | |
| 23 | 22 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴)) |
| 24 | ssequn1 4186 | . . . 4 ⊢ (( O ‘𝐴) ⊆ ( M ‘𝐴) ↔ (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴)) | |
| 25 | 23, 24 | sylib 218 | . . 3 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∪ ( M ‘𝐴)) = ( M ‘𝐴)) |
| 26 | 8, 21, 25 | 3eqtrrd 2782 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴)) |
| 27 | onsuc 7831 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
| 28 | oldval 27893 | . . 3 ⊢ (suc 𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴)) | |
| 29 | 27, 28 | syl 17 | . 2 ⊢ (𝐴 ∈ On → ( O ‘suc 𝐴) = ∪ ( M “ suc 𝐴)) |
| 30 | 26, 29 | eqtr4d 2780 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 “ cima 5688 Oncon0 6384 suc csuc 6386 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 No csur 27684 M cmade 27881 O cold 27882 |
| 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-pow 5365 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-rmo 3380 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-tp 4631 df-op 4633 df-uni 4908 df-int 4947 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 |
| This theorem is referenced by: oldsuc 27924 oldlim 27925 |
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