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Mirrors > Home > MPE Home > Th. List > ackbij1lem8 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
Ref | Expression |
---|---|
ackbij1lem8 | ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4187 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | fveq2d 6195 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴})) |
3 | pweq 4161 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴) | |
4 | 3 | fveq2d 6195 | . . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴)) |
5 | 2, 4 | eqeq12d 2637 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴))) |
6 | ackbij1lem4 9045 | . . . 4 ⊢ (𝑎 ∈ ω → {𝑎} ∈ (𝒫 ω ∩ Fin)) | |
7 | ackbij.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) | |
8 | 7 | ackbij1lem7 9048 | . . . 4 ⊢ ({𝑎} ∈ (𝒫 ω ∩ Fin) → (𝐹‘{𝑎}) = (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦))) |
10 | vex 3203 | . . . . . 6 ⊢ 𝑎 ∈ V | |
11 | sneq 4187 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → {𝑦} = {𝑎}) | |
12 | pweq 4161 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎) | |
13 | 11, 12 | xpeq12d 5140 | . . . . . 6 ⊢ (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)) |
14 | 10, 13 | iunxsn 4603 | . . . . 5 ⊢ ∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎) |
15 | 14 | fveq2i 6194 | . . . 4 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎)) |
16 | vpwex 4849 | . . . . . 6 ⊢ 𝒫 𝑎 ∈ V | |
17 | xpsnen2g 8053 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎) | |
18 | 10, 16, 17 | mp2an 708 | . . . . 5 ⊢ ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 |
19 | carden2b 8793 | . . . . 5 ⊢ (({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 → (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎)) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ (card‘({𝑎} × 𝒫 𝑎)) = (card‘𝒫 𝑎) |
21 | 15, 20 | eqtri 2644 | . . 3 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎) |
22 | 9, 21 | syl6eq 2672 | . 2 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎)) |
23 | 5, 22 | vtoclga 3272 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 𝒫 cpw 4158 {csn 4177 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 ωcom 7065 ≈ cen 7952 Fincfn 7955 cardccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-er 7742 df-en 7956 df-fin 7959 df-card 8765 |
This theorem is referenced by: ackbij1lem14 9055 ackbij1b 9061 |
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