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Mirrors > Home > MPE Home > Th. List > ackbij1lem8 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9662. (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 4579 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | fveq2d 6676 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹‘{𝑎}) = (𝐹‘{𝐴})) |
3 | pweq 4557 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴) | |
4 | 3 | fveq2d 6676 | . . 3 ⊢ (𝑎 = 𝐴 → (card‘𝒫 𝑎) = (card‘𝒫 𝐴)) |
5 | 2, 4 | eqeq12d 2839 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹‘{𝑎}) = (card‘𝒫 𝑎) ↔ (𝐹‘{𝐴}) = (card‘𝒫 𝐴))) |
6 | ackbij1lem4 9647 | . . . 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 9650 | . . . 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 3499 | . . . . . 6 ⊢ 𝑎 ∈ V | |
11 | sneq 4579 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → {𝑦} = {𝑎}) | |
12 | pweq 4557 | . . . . . . 7 ⊢ (𝑦 = 𝑎 → 𝒫 𝑦 = 𝒫 𝑎) | |
13 | 11, 12 | xpeq12d 5588 | . . . . . 6 ⊢ (𝑦 = 𝑎 → ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎)) |
14 | 10, 13 | iunxsn 5015 | . . . . 5 ⊢ ∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦) = ({𝑎} × 𝒫 𝑎) |
15 | 14 | fveq2i 6675 | . . . 4 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘({𝑎} × 𝒫 𝑎)) |
16 | vpwex 5280 | . . . . . 6 ⊢ 𝒫 𝑎 ∈ V | |
17 | xpsnen2g 8612 | . . . . . 6 ⊢ ((𝑎 ∈ V ∧ 𝒫 𝑎 ∈ V) → ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎) | |
18 | 10, 16, 17 | mp2an 690 | . . . . 5 ⊢ ({𝑎} × 𝒫 𝑎) ≈ 𝒫 𝑎 |
19 | carden2b 9398 | . . . . 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 2846 | . . 3 ⊢ (card‘∪ 𝑦 ∈ {𝑎} ({𝑦} × 𝒫 𝑦)) = (card‘𝒫 𝑎) |
22 | 9, 21 | syl6eq 2874 | . 2 ⊢ (𝑎 ∈ ω → (𝐹‘{𝑎}) = (card‘𝒫 𝑎)) |
23 | 5, 22 | vtoclga 3576 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 𝒫 cpw 4541 {csn 4569 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ‘cfv 6357 ωcom 7582 ≈ cen 8508 Fincfn 8511 cardccrd 9366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-en 8512 df-fin 8515 df-card 9370 |
This theorem is referenced by: ackbij1lem14 9657 ackbij1b 9663 |
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