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Mirrors > Home > MPE Home > Th. List > ackbij1lem10 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 9654. (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 4173 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ Fin) | |
3 | snfi 8588 | . . . . . 6 ⊢ {𝑦} ∈ Fin | |
4 | elinel1 4172 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ∈ 𝒫 ω) | |
5 | 4 | elpwid 4553 | . . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ ω) |
6 | onfin2 8704 | . . . . . . . . . 10 ⊢ ω = (On ∩ Fin) | |
7 | inss2 4206 | . . . . . . . . . 10 ⊢ (On ∩ Fin) ⊆ Fin | |
8 | 6, 7 | eqsstri 4001 | . . . . . . . . 9 ⊢ ω ⊆ Fin |
9 | 5, 8 | sstrdi 3979 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → 𝑥 ⊆ Fin) |
10 | 9 | sselda 3967 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ Fin) |
11 | pwfi 8813 | . . . . . . 7 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin) | |
12 | 10, 11 | sylib 220 | . . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝒫 𝑦 ∈ Fin) |
13 | xpfi 8783 | . . . . . 6 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
14 | 3, 12, 13 | sylancr 589 | . . . . 5 ⊢ ((𝑥 ∈ (𝒫 ω ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ({𝑦} × 𝒫 𝑦) ∈ Fin) |
15 | 14 | ralrimiva 3182 | . . . 4 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
16 | iunfi 8806 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) | |
17 | 2, 15, 16 | syl2anc 586 | . . 3 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) ∈ Fin) |
18 | ficardom 9384 | . . 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 6871 | 1 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∩ cin 3935 𝒫 cpw 4539 {csn 4561 ∪ ciun 4912 ↦ cmpt 5139 × cxp 5548 Oncon0 6186 ⟶wf 6346 ‘cfv 6350 ωcom 7574 Fincfn 8503 cardccrd 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 |
This theorem is referenced by: ackbij1lem12 9647 ackbij1lem13 9648 ackbij1lem14 9649 ackbij1lem15 9650 ackbij1lem16 9651 ackbij1lem17 9652 ackbij1lem18 9653 ackbij1b 9655 |
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