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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 1367 | . . . 4 ⊢ ⊤ | |
2 | csbeq1a 3078 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) | |
3 | 2 | equcoms 1718 | . . . . . 6 ⊢ (𝑧 = 𝑥 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
4 | trud 1379 | . . . . . 6 ⊢ (𝑧 = 𝑥 → ⊤) | |
5 | 3, 4 | 2thd 175 | . . . . 5 ⊢ (𝑧 = 𝑥 → (𝐵 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ⊤)) |
6 | 5 | rspcev 2853 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
7 | 1, 6 | mpan2 425 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
8 | elabrex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
9 | eqeq1 2194 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) | |
10 | 9 | rexbidv 2488 | . . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)) |
11 | 8, 10 | elab 2893 | . . 3 ⊢ (𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} ↔ ∃𝑧 ∈ 𝐴 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
12 | 7, 11 | sylibr 134 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵}) |
13 | nfv 1538 | . . . 4 ⊢ Ⅎ𝑧 𝑦 = 𝐵 | |
14 | nfcsb1v 3102 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 | |
15 | 14 | nfeq2 2341 | . . . 4 ⊢ Ⅎ𝑥 𝑦 = ⦋𝑧 / 𝑥⦌𝐵 |
16 | 2 | eqeq2d 2199 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑦 = ⦋𝑧 / 𝑥⦌𝐵)) |
17 | 13, 15, 16 | cbvrex 2712 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵) |
18 | 17 | abbii 2303 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = ⦋𝑧 / 𝑥⦌𝐵} |
19 | 12, 18 | eleqtrrdi 2281 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ⊤wtru 1364 ∈ wcel 2158 {cab 2173 ∃wrex 2466 Vcvv 2749 ⦋csb 3069 |
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 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rex 2471 df-v 2751 df-sbc 2975 df-csb 3070 |
This theorem is referenced by: eusvobj2 5874 |
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