Theorem oncardval 6812

Description: The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Assertion

Ref	Expression
oncardval	⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem oncardval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrefg 6479	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
2		breq1 3848	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	2	rspcev 2722	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
4	1, 3	mpdan 412	. 2 ⊢ (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
5		cardval3ex 6811	. 2 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
6	4, 5	syl 14	1 ⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})

Syntax hints:   → wi 4   = wceq 1289   ∈ wcel 1438  ∃wrex 2360  {crab 2363  ∩ cint 3688   class class class wbr 3845  Oncon0 4190  ‘cfv 5015   ≈ cen 6453  cardccrd 6805

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-en 6456  df-card 6806

This theorem is referenced by:  cardonle  6813