Theorem cardval3ex 6716

Description: The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
cardval3ex	⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem cardval3ex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		encv 6393	. . . 4 ⊢ (𝑥 ≈ 𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
2	1	simprd 112	. . 3 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
3	2	rexlimivw 2479	. 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
4		breq1 3814	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
5	4	cbvrexv 2584	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
6		intexrabim 3954	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
7	5, 6	sylbir 133	. 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
8		breq2 3815	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑦 ≈ 𝑧 ↔ 𝑦 ≈ 𝐴))
9	8	rabbidv 2601	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10	9	inteqd 3667	. . 3 ⊢ (𝑧 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
11		df-card 6711	. . 3 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
12	10, 11	fvmptg 5325	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
13	3, 7, 12	syl2anc 403	1 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})

Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  ∃wrex 2354  {crab 2357  Vcvv 2612  ∩ cint 3662   class class class wbr 3811  Oncon0 4154  ‘cfv 4969   ≈ cen 6385  cardccrd 6710

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fv 4977  df-en 6388  df-card 6711

This theorem is referenced by:  oncardval  6717  carden2bex  6720