Theorem cardval3ex 7186

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 6748	. . . 4 ⊢ (𝑥 ≈ 𝐴 → (𝑥 ∈ V ∧ 𝐴 ∈ V))
2	1	simprd 114	. . 3 ⊢ (𝑥 ≈ 𝐴 → 𝐴 ∈ V)
3	2	rexlimivw 2590	. 2 ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → 𝐴 ∈ V)
4		breq1 4008	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
5	4	cbvrexv 2706	. . 3 ⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
6		intexrabim 4155	. . 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 4009	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑦 ≈ 𝑧 ↔ 𝑦 ≈ 𝐴))
9	8	rabbidv 2728	. . . 4 ⊢ (𝑧 = 𝐴 → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
10	9	inteqd 3851	. . 3 ⊢ (𝑧 = 𝐴 → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
11		df-card 7181	. . 3 ⊢ card = (𝑧 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})
12	10, 11	fvmptg 5594	. 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 1353   ∈ wcel 2148  ∃wrex 2456  {crab 2459  Vcvv 2739  ∩ cint 3846   class class class wbr 4005  Oncon0 4365  ‘cfv 5218   ≈ cen 6740  cardccrd 7180

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-en 6743  df-card 7181

This theorem is referenced by:  oncardval  7187  carden2bex  7190