![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cantnfdm | Structured version Visualization version GIF version |
Description: The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
Ref | Expression |
---|---|
cantnffval.s | ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅} |
cantnffval.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnffval.b | ⊢ (𝜑 → 𝐵 ∈ On) |
Ref | Expression |
---|---|
cantnfdm | ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnffval.s | . . . 4 ⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑔 finSupp ∅} | |
2 | cantnffval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ On) | |
3 | cantnffval.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ On) | |
4 | 1, 2, 3 | cantnffval 8837 | . . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) |
5 | 4 | dmeqd 5558 | . 2 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ))) |
6 | fvex 6446 | . . . . 5 ⊢ (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V | |
7 | 6 | csbex 5018 | . . . 4 ⊢ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V |
8 | 7 | rgenw 3133 | . . 3 ⊢ ∀𝑓 ∈ 𝑆 ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V |
9 | dmmptg 5873 | . . 3 ⊢ (∀𝑓 ∈ 𝑆 ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ) ∈ V → dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)) = 𝑆) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ dom (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)) = 𝑆 |
11 | 5, 10 | syl6eq 2877 | 1 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3117 {crab 3121 Vcvv 3414 ⦋csb 3757 ∅c0 4144 class class class wbr 4873 ↦ cmpt 4952 E cep 5254 dom cdm 5342 Oncon0 5963 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 supp csupp 7559 seq𝜔cseqom 7808 +o coa 7823 ·o comu 7824 ↑o coe 7825 ↑𝑚 cmap 8122 finSupp cfsupp 8544 OrdIsocoi 8683 CNF ccnf 8835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-seqom 7809 df-cnf 8836 |
This theorem is referenced by: cantnfs 8840 cantnfval 8842 cantnff 8848 oemapso 8856 wemapwe 8871 oef1o 8872 |
Copyright terms: Public domain | W3C validator |