Theorem cardid2 9877

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 9876	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
2		ssrab2 4020	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On
3		fvex 6853	. . . . . 6 ⊢ (card‘𝐴) ∈ V
4	1, 3	eqeltrrdi 2845	. . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
5		intex 5285	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
6	4, 5	sylibr 234	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅)
7		onint 7744	. . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
8	2, 6, 7	sylancr 588	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	1, 8	eqeltrd 2836	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10		breq1 5088	. . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
11	10	elrab 3634	. . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
12	11	simprbi 497	. 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 2932  {crab 3389  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  ∩ cint 4889   class class class wbr 5085  dom cdm 5631  Oncon0 6323  ‘cfv 6498   ≈ cen 8890  cardccrd 9859

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-en 8894  df-card 9863

This theorem is referenced by:  isnum3  9878  oncardid  9880  cardidm  9883  ficardom  9885  ficardid  9886  cardnn  9887  cardnueq0  9888  carden2a  9890  carden2b  9891  carddomi2  9894  sdomsdomcardi  9895  cardsdomelir  9897  cardsdomel  9898  infxpidm2  9939  dfac8b  9953  numdom  9960  alephnbtwn2  9994  alephsucdom  10001  infenaleph  10013  dfac12r  10069  cardadju  10117  pwsdompw  10125  cff1  10180  cfflb  10181  cflim2  10185  cfss  10187  cfslb  10188  domtriomlem  10364  cardid  10469  cardidg  10470  carden  10473  sdomsdomcard  10482  hargch  10596  gch2  10598  hashkf  14294