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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 9371 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | 1 | ensymd 8549 | . . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴)) |
3 | entr 8550 | . . . . . . . 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 8550 | . . . . . . . 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 213 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ (card‘𝐴))) |
10 | 9 | rabbidv 3481 | . . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
11 | 10 | inteqd 4874 | . . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
12 | cardval3 9370 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
13 | cardon 9362 | . . . 4 ⊢ (card‘𝐴) ∈ On | |
14 | oncardval 9373 | . . . 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 2867 | . 2 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
17 | card0 9376 | . . 3 ⊢ (card‘∅) = ∅ | |
18 | ndmfv 6694 | . . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
19 | 18 | fveq2d 6668 | . . 3 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅)) |
20 | 17, 19, 18 | 3eqtr4a 2882 | . 2 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
21 | 16, 20 | pm2.61i 183 | 1 ⊢ (card‘(card‘𝐴)) = (card‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3142 ∅c0 4290 ∩ cint 4869 class class class wbr 5058 dom cdm 5549 Oncon0 6185 ‘cfv 6349 ≈ cen 8495 cardccrd 9353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-card 9357 |
This theorem is referenced by: oncard 9378 cardlim 9390 cardiun 9400 alephnbtwn2 9487 infenaleph 9506 dfac12k 9562 pwsdompw 9615 cardcf 9663 cfeq0 9667 cfflb 9670 alephval2 9983 cfpwsdom 9995 gch2 10086 tskcard 10192 hashcard 13706 iscard4 39780 |
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