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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 6392 | . . . . . . . 8 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
2 | 1 | imaeq2i 6078 | . . . . . . 7 ⊢ ( M “ suc 𝐴) = ( M “ (𝐴 ∪ {𝐴})) |
3 | imaundi 6172 | . . . . . . 7 ⊢ ( M “ (𝐴 ∪ {𝐴})) = (( M “ 𝐴) ∪ ( M “ {𝐴})) | |
4 | 2, 3 | eqtri 2763 | . . . . . 6 ⊢ ( M “ suc 𝐴) = (( M “ 𝐴) ∪ ( M “ {𝐴})) |
5 | 4 | unieqi 4924 | . . . . 5 ⊢ ∪ ( M “ suc 𝐴) = ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) |
6 | uniun 4935 | . . . . 5 ⊢ ∪ (( M “ 𝐴) ∪ ( M “ {𝐴})) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) | |
7 | 5, 6 | eqtri 2763 | . . . 4 ⊢ ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) |
8 | 7 | a1i 11 | . . 3 ⊢ (𝐴 ∈ On → ∪ ( M “ suc 𝐴) = (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴}))) |
9 | oldval 27908 | . . . . 5 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) | |
10 | 9 | eqcomd 2741 | . . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) = ( O ‘𝐴)) |
11 | madef 27910 | . . . . . . . 8 ⊢ M :On⟶𝒫 No | |
12 | ffn 6737 | . . . . . . . 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 2741 | . . . . . . 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 4925 | . . . . 5 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ∪ {( M ‘𝐴)}) |
18 | fvex 6920 | . . . . . 6 ⊢ ( M ‘𝐴) ∈ V | |
19 | 18 | unisn 4931 | . . . . 5 ⊢ ∪ {( M ‘𝐴)} = ( M ‘𝐴) |
20 | 17, 19 | eqtrdi 2791 | . . . 4 ⊢ (𝐴 ∈ On → ∪ ( M “ {𝐴}) = ( M ‘𝐴)) |
21 | 10, 20 | uneq12d 4179 | . . 3 ⊢ (𝐴 ∈ On → (∪ ( M “ 𝐴) ∪ ∪ ( M “ {𝐴})) = (( O ‘𝐴) ∪ ( M ‘𝐴))) |
22 | oldssmade 27931 | . . . . 5 ⊢ ( O ‘𝐴) ⊆ ( M ‘𝐴) | |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ On → ( O ‘𝐴) ⊆ ( M ‘𝐴)) |
24 | ssequn1 4196 | . . . 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 2780 | . 2 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ∪ ( M “ suc 𝐴)) |
27 | onsuc 7831 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
28 | oldval 27908 | . . 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 2778 | 1 ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 ∪ cuni 4912 “ cima 5692 Oncon0 6386 suc csuc 6388 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 No csur 27699 M cmade 27896 O cold 27897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 df-scut 27843 df-made 27901 df-old 27902 |
This theorem is referenced by: oldsuc 27939 oldlim 27940 |
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