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Mirrors > Home > MPE Home > Th. List > cardidm | Structured version Visualization version GIF version |
Description: The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
Ref | Expression |
---|---|
cardidm | ⊢ (card‘(card‘𝐴)) = (card‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9711 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | 1 | ensymd 8791 | . . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴)) |
3 | entr 8792 | . . . . . . . 8 ⊢ ((𝑦 ≈ 𝐴 ∧ 𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴)) | |
4 | 3 | expcom 414 | . . . . . . 7 ⊢ (𝐴 ≈ (card‘𝐴) → (𝑦 ≈ 𝐴 → 𝑦 ≈ (card‘𝐴))) |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 → 𝑦 ≈ (card‘𝐴))) |
6 | entr 8792 | . . . . . . . 8 ⊢ ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦 ≈ 𝐴) | |
7 | 6 | expcom 414 | . . . . . . 7 ⊢ ((card‘𝐴) ≈ 𝐴 → (𝑦 ≈ (card‘𝐴) → 𝑦 ≈ 𝐴)) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ (card‘𝐴) → 𝑦 ≈ 𝐴)) |
9 | 5, 8 | impbid 211 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ (card‘𝐴))) |
10 | 9 | rabbidv 3414 | . . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
11 | 10 | inteqd 4884 | . . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
12 | cardval3 9710 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
13 | cardon 9702 | . . . 4 ⊢ (card‘𝐴) ∈ On | |
14 | oncardval 9713 | . . . 4 ⊢ ((card‘𝐴) ∈ On → (card‘(card‘𝐴)) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) | |
15 | 13, 14 | mp1i 13 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
16 | 11, 12, 15 | 3eqtr4rd 2789 | . 2 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
17 | card0 9716 | . . 3 ⊢ (card‘∅) = ∅ | |
18 | ndmfv 6804 | . . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
19 | 18 | fveq2d 6778 | . . 3 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅)) |
20 | 17, 19, 18 | 3eqtr4a 2804 | . 2 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
21 | 16, 20 | pm2.61i 182 | 1 ⊢ (card‘(card‘𝐴)) = (card‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2106 {crab 3068 ∅c0 4256 ∩ cint 4879 class class class wbr 5074 dom cdm 5589 Oncon0 6266 ‘cfv 6433 ≈ cen 8730 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-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
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-rab 3073 df-v 3434 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-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-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-ord 6269 df-on 6270 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-er 8498 df-en 8734 df-card 9697 |
This theorem is referenced by: oncard 9718 cardlim 9730 cardiun 9740 alephnbtwn2 9828 infenaleph 9847 dfac12k 9903 pwsdompw 9960 cardcf 10008 cfeq0 10012 cfflb 10015 alephval2 10328 cfpwsdom 10340 gch2 10431 tskcard 10537 hashcard 14070 iscard4 41140 |
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