Theorem cardid2 9366

Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Assertion

Ref	Expression
cardid2	⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)

Proof of Theorem cardid2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardval3 9365	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
2		ssrab2 4007	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On
3		fvex 6658	. . . . . 6 ⊢ (card‘𝐴) ∈ V
4	1, 3	eqeltrrdi 2899	. . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
5		intex 5204	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
6	4, 5	sylibr 237	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅)
7		onint 7490	. . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
8	2, 6, 7	sylancr 590	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	1, 8	eqeltrd 2890	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10		breq1 5033	. . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
11	10	elrab 3628	. . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
12	11	simprbi 500	. 2 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} → (card‘𝐴) ≈ 𝐴)
13	9, 12	syl 17	1 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838   class class class wbr 5030  dom cdm 5519  Oncon0 6159  ‘cfv 6324   ≈ cen 8489  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-en 8493  df-card 9352

This theorem is referenced by:  isnum3  9367  oncardid  9369  cardidm  9372  ficardom  9374  ficardid  9375  cardnn  9376  cardnueq0  9377  carden2a  9379  carden2b  9380  carddomi2  9383  sdomsdomcardi  9384  cardsdomelir  9386  cardsdomel  9387  infxpidm2  9428  dfac8b  9442  numdom  9449  alephnbtwn2  9483  alephsucdom  9490  infenaleph  9502  dfac12r  9557  cardadju  9605  pwsdompw  9615  cff1  9669  cfflb  9670  cflim2  9674  cfss  9676  cfslb  9677  domtriomlem  9853  cardid  9958  cardidg  9959  carden  9962  sdomsdomcard  9971  hargch  10084  gch2  10086  hashkf  13688