Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9562 | . . . . . . . 8 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴) | |
2 | relen 8560 | . . . . . . . . 9 ⊢ Rel ≈ | |
3 | 2 | brrelex1i 5579 | . . . . . . . 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 4022 | . . . . . . 7 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) | |
6 | ssdomg 8601 | . . . . . . 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 8614 | . . . . . 6 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≈ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) | |
9 | 7, 1, 8 | syl2anc 587 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) |
10 | numdom 9538 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card) | |
11 | 9, 10 | syldan 594 | . . . 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 8969 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) |
14 | 13 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑥):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) |
15 | fodomnum 9557 | . . . 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 8608 | . . 3 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴) | |
18 | 16, 9, 17 | syl2anc 587 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≼ 𝐴) |
19 | fvex 6687 | . . 3 ⊢ (fi‘𝐴) ∈ V | |
20 | ssfii 8956 | . . . 4 ⊢ (𝐴 ∈ dom card → 𝐴 ⊆ (fi‘𝐴)) | |
21 | 20 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → 𝐴 ⊆ (fi‘𝐴)) |
22 | ssdomg 8601 | . . 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 8687 | . 2 ⊢ (((fi‘𝐴) ≼ 𝐴 ∧ 𝐴 ≼ (fi‘𝐴)) → (fi‘𝐴) ≈ 𝐴) | |
25 | 18, 23, 24 | syl2anc 587 | 1 ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 Vcvv 3398 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 {csn 4516 ∩ cint 4836 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5525 –onto→wfo 6337 ‘cfv 6339 ωcom 7599 ≈ cen 8552 ≼ cdom 8553 Fincfn 8555 ficfi 8947 cardccrd 9437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-seqom 8113 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fi 8948 df-oi 9047 df-card 9441 df-acn 9444 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |