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Mirrors > Home > MPE Home > Th. List > ackbij1lem7 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) |
Ref | Expression |
---|---|
ackbij1lem7 | ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 5016 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦) = ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) | |
2 | 1 | fveq2d 6904 | . 2 ⊢ (𝑥 = 𝐴 → (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))) |
3 | ackbij.f | . 2 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) | |
4 | fvex 6913 | . 2 ⊢ (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ V | |
5 | 2, 3, 4 | fvmpt 7008 | 1 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3945 𝒫 cpw 4606 {csn 4632 ∪ ciun 5000 ↦ cmpt 5235 × cxp 5679 ‘cfv 6553 ωcom 7875 Fincfn 8973 cardccrd 9974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-iota 6505 df-fun 6555 df-fv 6561 |
This theorem is referenced by: ackbij1lem8 10266 ackbij1lem9 10267 |
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