![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ackbij1lem12 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10239. (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 |
---|---|
ackbij1lem12 | ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ackbij.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) | |
2 | 1 | ackbij1lem10 10230 | . . . 4 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω |
3 | 1 | ackbij1lem11 10231 | . . . 4 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ (𝒫 ω ∩ Fin)) |
4 | ffvelcdm 7083 | . . . 4 ⊢ ((𝐹:(𝒫 ω ∩ Fin)⟶ω ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) → (𝐹‘𝐴) ∈ ω) | |
5 | 2, 3, 4 | sylancr 586 | . . 3 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ∈ ω) |
6 | difssd 4132 | . . . . 5 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∖ 𝐴) ⊆ 𝐵) | |
7 | 1 | ackbij1lem11 10231 | . . . . 5 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐵 ∖ 𝐴) ⊆ 𝐵) → (𝐵 ∖ 𝐴) ∈ (𝒫 ω ∩ Fin)) |
8 | 6, 7 | syldan 590 | . . . 4 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∖ 𝐴) ∈ (𝒫 ω ∩ Fin)) |
9 | ffvelcdm 7083 | . . . 4 ⊢ ((𝐹:(𝒫 ω ∩ Fin)⟶ω ∧ (𝐵 ∖ 𝐴) ∈ (𝒫 ω ∩ Fin)) → (𝐹‘(𝐵 ∖ 𝐴)) ∈ ω) | |
10 | 2, 8, 9 | sylancr 586 | . . 3 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘(𝐵 ∖ 𝐴)) ∈ ω) |
11 | nnaword1 8635 | . . 3 ⊢ (((𝐹‘𝐴) ∈ ω ∧ (𝐹‘(𝐵 ∖ 𝐴)) ∈ ω) → (𝐹‘𝐴) ⊆ ((𝐹‘𝐴) +o (𝐹‘(𝐵 ∖ 𝐴)))) | |
12 | 5, 10, 11 | syl2anc 583 | . 2 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ ((𝐹‘𝐴) +o (𝐹‘(𝐵 ∖ 𝐴)))) |
13 | disjdif 4471 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
15 | 1 | ackbij1lem9 10229 | . . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐵 ∖ 𝐴) ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) → (𝐹‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝐹‘𝐴) +o (𝐹‘(𝐵 ∖ 𝐴)))) |
16 | 3, 8, 14, 15 | syl3anc 1370 | . . 3 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝐹‘𝐴) +o (𝐹‘(𝐵 ∖ 𝐴)))) |
17 | undif 4481 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
18 | 17 | biimpi 215 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
19 | 18 | adantl 481 | . . . 4 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
20 | 19 | fveq2d 6895 | . . 3 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘(𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝐹‘𝐵)) |
21 | 16, 20 | eqtr3d 2773 | . 2 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → ((𝐹‘𝐴) +o (𝐹‘(𝐵 ∖ 𝐴))) = (𝐹‘𝐵)) |
22 | 12, 21 | sseqtrd 4022 | 1 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ ciun 4997 ↦ cmpt 5231 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ωcom 7859 +o coa 8469 Fincfn 8945 cardccrd 9936 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 |
This theorem is referenced by: ackbij1lem15 10235 ackbij1b 10240 |
Copyright terms: Public domain | W3C validator |