Theorem cardid2 9868

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 9867	. . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
2		ssrab2 4011	. . . 4 ⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On
3		fvex 6840	. . . . . 6 ⊢ (card‘𝐴) ∈ V
4	1, 3	eqeltrrdi 2848	. . . . 5 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
5		intex 5272	. . . . 5 ⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅ ↔ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
6	4, 5	sylibr 235	. . . 4 ⊢ (𝐴 ∈ dom card → {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅)
7		onint 7733	. . . 4 ⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
8	2, 6, 7	sylancr 593	. . 3 ⊢ (𝐴 ∈ dom card → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
9	1, 8	eqeltrd 2839	. 2 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10		breq1 5075	. . . 4 ⊢ (𝑦 = (card‘𝐴) → (𝑦 ≈ 𝐴 ↔ (card‘𝐴) ≈ 𝐴))
11	10	elrab 3629	. . 3 ⊢ ((card‘𝐴) ∈ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ↔ ((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴))
12	11	simprbi 498	. 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 2119   ≠ wne 2934  {crab 3391  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  ∩ cint 4877   class class class wbr 5072  dom cdm 5618  Oncon0 6310  ‘cfv 6485   ≈ cen 8880  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-en 8884  df-card 9854

This theorem is referenced by:  isnum3  9869  oncardid  9871  cardidm  9874  ficardom  9876  ficardid  9877  cardnn  9878  cardnueq0  9879  carden2a  9881  carden2b  9882  carddomi2  9885  sdomsdomcardi  9886  cardsdomelir  9888  cardsdomel  9889  infxpidm2  9930  dfac8b  9944  numdom  9951  alephnbtwn2  9985  alephsucdom  9992  infenaleph  10004  dfac12r  10060  cardadju  10108  pwsdompw  10116  cff1  10171  cfflb  10172  cflim2  10176  cfss  10178  cfslb  10179  domtriomlem  10355  cardid  10460  cardidg  10461  carden  10464  sdomsdomcard  10473  hargch  10587  gch2  10589  hashkf  14285