| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ackbij1lem8 | Structured version Visualization version GIF version | ||
| Description: Lemma for ackbij1 10277. (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 4636 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
| 2 | 1 | fveq2d 6910 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴})) |
| 3 | pweq 4614 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴) | |
| 4 | 3 | fveq2d 6910 | . . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴)) |
| 5 | 2, 4 | eqeq12d 2753 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴))) |
| 6 | ackbij1lem4 10262 | . . . 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 10265 | . . . 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 3484 | . . . . . 6 ⊢ 𝑎 ∈ V | |
| 11 | sneq 4636 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → {𝑦} = {𝑎}) | |
| 12 | pweq 4614 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎) | |
| 13 | 11, 12 | xpeq12d 5716 | . . . . . 6 ⊢ (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)) |
| 14 | 10, 13 | iunxsn 5091 | . . . . 5 ⊢ ∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎) |
| 15 | 14 | fveq2i 6909 | . . . 4 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎)) |
| 16 | vpwex 5377 | . . . . . 6 ⊢ 𝒫 𝑎 ∈ V | |
| 17 | xpsnen2g 9105 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎) | |
| 18 | 10, 16, 17 | mp2an 692 | . . . . 5 ⊢ ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 |
| 19 | carden2b 10007 | . . . . 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 2765 | . . 3 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎) |
| 22 | 9, 21 | eqtrdi 2793 | . 2 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎)) |
| 23 | 5, 22 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 𝒫 cpw 4600 {csn 4626 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 ωcom 7887 ≈ cen 8982 Fincfn 8985 cardccrd 9975 |
| 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-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-rab 3437 df-v 3482 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-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-ord 6387 df-on 6388 df-lim 6389 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-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-er 8745 df-en 8986 df-fin 8989 df-card 9979 |
| This theorem is referenced by: ackbij1lem14 10272 ackbij1b 10278 |
| Copyright terms: Public domain | W3C validator |