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Mirrors > Home > MPE Home > Th. List > inffien | Structured version Visualization version GIF version |
Description: The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.) |
Ref | Expression |
---|---|
inffien | ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infpwfien 9818 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴) | |
2 | relen 8738 | . . . . . . . . 9 ⊢ Rel ≈ | |
3 | 2 | brrelex1i 5643 | . . . . . . . 8 ⊢ ((𝒫 𝐴 ∩ Fin) ≈ 𝐴 → (𝒫 𝐴 ∩ Fin) ∈ V) |
4 | 1, 3 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V) |
5 | difss 4066 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) | |
6 | ssdomg 8786 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin))) | |
7 | 4, 5, 6 | mpisyl 21 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin)) |
8 | domentr 8799 | . . . . . 6 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≈ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) | |
9 | 7, 1, 8 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) |
10 | numdom 9794 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card) | |
11 | 9, 10 | syldan 591 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card) |
12 | eqid 2738 | . . . . . 6 ⊢ (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥) = (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥) | |
13 | 12 | fifo 9191 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) |
15 | fodomnum 9813 | . . . 4 ⊢ (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card → ((𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}))) | |
16 | 11, 14, 15 | sylc 65 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅})) |
17 | domtr 8793 | . . 3 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴) | |
18 | 16, 9, 17 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴) |
19 | fvex 6787 | . . 3 ⊢ (fi‘𝐴) ∈ V | |
20 | ssfii 9178 | . . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ⊆ (fi‘𝐴)) | |
21 | 20 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ⊆ (fi‘𝐴)) |
22 | ssdomg 8786 | . . 3 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴))) | |
23 | 19, 21, 22 | mpsyl 68 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ≼ (fi‘𝐴)) |
24 | sbth 8880 | . 2 ⊢ (((fi‘𝐴) ≼ 𝐴 ∧ 𝐴 ≼ (fi‘𝐴)) → (fi‘𝐴) ≈ 𝐴) | |
25 | 18, 23, 24 | syl2anc 584 | 1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3432 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ∩ cint 4879 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 –onto→wfo 6431 ‘cfv 6433 ωcom 7712 ≈ cen 8730 ≼ cdom 8731 Fincfn 8733 ficfi 9169 cardccrd 9693 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-seqom 8279 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-oi 9269 df-card 9697 df-acn 9700 |
This theorem is referenced by: (None) |
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