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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9852. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
Ref | Expression |
---|---|
ackbij1lem10 | ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackbij.f | . 2 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) | |
2 | elinel2 4110 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ Fin) | |
3 | snfi 8721 | . . . . . 6 ⊢ {𝑦} ∈ Fin | |
4 | elinel1 4109 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ 𝒫 ω) | |
5 | 4 | elpwid 4524 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ ω) |
6 | onfin2 8871 | . . . . . . . . . 10 ⊢ ω = (On ∩ Fin) | |
7 | inss2 4144 | . . . . . . . . . 10 ⊢ (On ∩ Fin) ⊆ Fin | |
8 | 6, 7 | eqsstri 3935 | . . . . . . . . 9 ⊢ ω ⊆ Fin |
9 | 5, 8 | sstrdi 3913 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ Fin) |
10 | 9 | sselda 3901 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ Fin) |
11 | pwfi 8856 | . . . . . . 7 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin) | |
12 | 10, 11 | sylib 221 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 𝑦 ∈ Fin) |
13 | xpfi 8942 | . . . . . 6 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
14 | 3, 12, 13 | sylancr 590 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ({𝑦} × 𝒫 𝑦) ∈ Fin) |
15 | 14 | ralrimiva 3105 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
16 | iunfi 8964 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
17 | 2, 15, 16 | syl2anc 587 | . . 3 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
18 | ficardom 9577 | . . 3 ⊢ (∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) ∈ ω) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) ∈ ω) |
20 | 1, 19 | fmpti 6929 | 1 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∩ cin 3865 𝒫 cpw 4513 {csn 4541 ∪ ciun 4904 ↦ cmpt 5135 × cxp 5549 Oncon0 6213 ⟶wf 6376 ‘cfv 6380 ωcom 7644 Fincfn 8626 cardccrd 9551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-om 7645 df-1st 7761 df-2nd 7762 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 |
This theorem is referenced by: ackbij1lem12 9845 ackbij1lem13 9846 ackbij1lem14 9847 ackbij1lem15 9848 ackbij1lem16 9849 ackbij1lem17 9850 ackbij1lem18 9851 ackbij1b 9853 |
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