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Mirrors > Home > ILE Home > Th. List > elabrex | GIF version |
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Ref | Expression |
---|---|
elabrex.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elabrex | ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1335 | . . . 4 ⊢ ⊤ | |
2 | csbeq1a 3012 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
3 | 2 | equcoms 1684 | . . . . . 6 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
4 | a1tru 1347 | . . . . . 6 ⊢ (𝑧 = 𝑥 → ⊤) | |
5 | 3, 4 | 2thd 174 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤)) |
6 | 5 | rspcev 2789 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
7 | 1, 6 | mpan2 421 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
8 | elabrex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
9 | eqeq1 2146 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) | |
10 | 9 | rexbidv 2438 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) |
11 | 8, 10 | elab 2828 | . . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
12 | 7, 11 | sylibr 133 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}) |
13 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵 | |
14 | nfcsb1v 3035 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 | |
15 | 14 | nfeq2 2293 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 |
16 | 2 | eqeq2d 2151 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)) |
17 | 13, 15, 16 | cbvrex 2651 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) |
18 | 17 | abbii 2255 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} |
19 | 12, 18 | eleqtrrdi 2233 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 {cab 2125 ∃wrex 2417 Vcvv 2686 ⦋csb 3003 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 |
This theorem is referenced by: eusvobj2 5760 |
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