Theorem oncardval 7163

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 6742	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ≈ 𝐴)
2		breq1 3992	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ≈ 𝐴 ↔ 𝐴 ≈ 𝐴))
3	2	rspcev 2834	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
4	1, 3	mpdan 419	. 2 ⊢ (𝐴 ∈ On → ∃𝑦 ∈ On 𝑦 ≈ 𝐴)
5		cardval3ex 7162	. 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 1348   ∈ wcel 2141  ∃wrex 2449  {crab 2452  ∩ cint 3831   class class class wbr 3989  Oncon0 4348  ‘cfv 5198   ≈ cen 6716  cardccrd 7156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-en 6719  df-card 7157

This theorem is referenced by:  cardonle  7164