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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 9374 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | 1 | ensymd 8552 | . . . . . . 7 ⊢ (𝐴 ∈ dom card → 𝐴 ≈ (card‘𝐴)) |
3 | entr 8553 | . . . . . . . 8 ⊢ ((𝑦 ≈ 𝐴 ∧ 𝐴 ≈ (card‘𝐴)) → 𝑦 ≈ (card‘𝐴)) | |
4 | 3 | expcom 416 | . . . . . . 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 8553 | . . . . . . . 8 ⊢ ((𝑦 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐴) → 𝑦 ≈ 𝐴) | |
7 | 6 | expcom 416 | . . . . . . 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 214 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝑦 ≈ 𝐴 ↔ 𝑦 ≈ (card‘𝐴))) |
10 | 9 | rabbidv 3479 | . . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
11 | 10 | inteqd 4872 | . . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ (card‘𝐴)}) |
12 | cardval3 9373 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | |
13 | cardon 9365 | . . . 4 ⊢ (card‘𝐴) ∈ On | |
14 | oncardval 9376 | . . . 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 2865 | . 2 ⊢ (𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
17 | card0 9379 | . . 3 ⊢ (card‘∅) = ∅ | |
18 | ndmfv 6693 | . . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅) | |
19 | 18 | fveq2d 6667 | . . 3 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘∅)) |
20 | 17, 19, 18 | 3eqtr4a 2880 | . 2 ⊢ (¬ 𝐴 ∈ dom card → (card‘(card‘𝐴)) = (card‘𝐴)) |
21 | 16, 20 | pm2.61i 184 | 1 ⊢ (card‘(card‘𝐴)) = (card‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1531 ∈ wcel 2108 {crab 3140 ∅c0 4289 ∩ cint 4867 class class class wbr 5057 dom cdm 5548 Oncon0 6184 ‘cfv 6348 ≈ cen 8498 cardccrd 9356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8281 df-en 8502 df-card 9360 |
This theorem is referenced by: oncard 9381 cardlim 9393 cardiun 9403 alephnbtwn2 9490 infenaleph 9509 dfac12k 9565 pwsdompw 9618 cardcf 9666 cfeq0 9670 cfflb 9673 alephval2 9986 cfpwsdom 9998 gch2 10089 tskcard 10195 hashcard 13708 iscard4 39891 |
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