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Mirrors > Home > ILE Home > Th. List > cardval3ex | GIF version |
Description: The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
cardval3ex | ⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) |
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}) |
Colors of variables: wff set class |
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 |
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