Theorem cardid2 9384

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 9383	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
2		ssrab2 4058	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On
3		fvex 6685	. . . . . 6 ⊢ (card‘𝐴) ∈ V
4	1, 3	eqeltrrdi 2924	. . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
5		intex 5242	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
6	4, 5	sylibr 236	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅)
7		onint 7512	. . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
8	2, 6, 7	sylancr 589	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	1, 8	eqeltrd 2915	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10		breq1 5071	. . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
11	10	elrab 3682	. . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
12	11	simprbi 499	. 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 2114   ≠ wne 3018  {crab 3144  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  ∩ cint 4878   class class class wbr 5068  dom cdm 5557  Oncon0 6193  ‘cfv 6357   ≈ cen 8508  cardccrd 9366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-en 8512  df-card 9370

This theorem is referenced by:  isnum3  9385  oncardid  9387  cardidm  9390  ficardom  9392  ficardid  9393  cardnn  9394  cardnueq0  9395  carden2a  9397  carden2b  9398  carddomi2  9401  sdomsdomcardi  9402  cardsdomelir  9404  cardsdomel  9405  infxpidm2  9445  dfac8b  9459  numdom  9466  alephnbtwn2  9500  alephsucdom  9507  infenaleph  9519  dfac12r  9574  cardadju  9622  pwsdompw  9628  cff1  9682  cfflb  9683  cflim2  9687  cfss  9689  cfslb  9690  domtriomlem  9866  cardid  9971  cardidg  9972  carden  9975  sdomsdomcard  9984  hargch  10097  gch2  10099  hashkf  13695