Theorem cardval3ex 7318

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 6856	. . . 4 ⊢ (𝑥 ≈ 𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
2	1	simprd 114	. . 3 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
3	2	rexlimivw 2621	. 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
4		breq1 4062	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
5	4	cbvrexv 2743	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
6		intexrabim 4213	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
7	5, 6	sylbir 135	. 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V)
8		breq2 4063	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑦 ≈ 𝑧 ↔ 𝑦 ≈ 𝐴))
9	8	rabbidv 2765	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10	9	inteqd 3904	. . 3 ⊢ (𝑧 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
11		df-card 7312	. . 3 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
12	10, 11	fvmptg 5678	. 2 ⊢ ((𝐴 ∈ V ∧ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴} ∈ V) → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
13	3, 7, 12	syl2anc 411	1 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  ∃wrex 2487  {crab 2490  Vcvv 2776  ∩ cint 3899   class class class wbr 4059  Oncon0 4428  ‘cfv 5290   ≈ cen 6848  cardccrd 7310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-en 6851  df-card 7312

This theorem is referenced by:  oncardval  7319  carden2bex  7323