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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10220. (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 4194 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ Fin) | |
3 | snfi 9032 | . . . . . 6 ⊢ {𝑦} ∈ Fin | |
4 | elinel1 4193 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ 𝒫 ω) | |
5 | 4 | elpwid 4607 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ ω) |
6 | onfin2 9219 | . . . . . . . . . 10 ⊢ ω = (On ∩ Fin) | |
7 | inss2 4227 | . . . . . . . . . 10 ⊢ (On ∩ Fin) ⊆ Fin | |
8 | 6, 7 | eqsstri 4014 | . . . . . . . . 9 ⊢ ω ⊆ Fin |
9 | 5, 8 | sstrdi 3992 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ Fin) |
10 | 9 | sselda 3980 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ Fin) |
11 | pwfi 9166 | . . . . . . 7 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin) | |
12 | 10, 11 | sylib 217 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 𝑦 ∈ Fin) |
13 | xpfi 9305 | . . . . . 6 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
14 | 3, 12, 13 | sylancr 588 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ({𝑦} × 𝒫 𝑦) ∈ Fin) |
15 | 14 | ralrimiva 3147 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
16 | iunfi 9328 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
17 | 2, 15, 16 | syl2anc 585 | . . 3 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
18 | ficardom 9943 | . . 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 7099 | 1 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3945 𝒫 cpw 4598 {csn 4624 ∪ ciun 4993 ↦ cmpt 5227 × cxp 5670 Oncon0 6356 ⟶wf 6531 ‘cfv 6535 ωcom 7842 Fincfn 8927 cardccrd 9917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-om 7843 df-1o 8453 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9921 |
This theorem is referenced by: ackbij1lem12 10213 ackbij1lem13 10214 ackbij1lem14 10215 ackbij1lem15 10216 ackbij1lem16 10217 ackbij1lem17 10218 ackbij1lem18 10219 ackbij1b 10221 |
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