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Mirrors > Home > ILE Home > Th. List > dfdmf | GIF version |
Description: Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
dfdmf.1 | ⊢ Ⅎ𝑥𝐴 |
dfdmf.2 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
dfdmf | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 4654 | . 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} | |
2 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
3 | dfdmf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 4064 | . . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 4022 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvex 1767 | . . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦) |
9 | 8 | abbii 2305 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} |
10 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
11 | dfdmf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
13 | 10, 11, 12 | nfbr 4064 | . . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
14 | 13 | nfex 1648 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦 |
15 | nfv 1539 | . . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦 | |
16 | breq1 4021 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1836 | . . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvab 2313 | . 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2214 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∃wex 1503 {cab 2175 Ⅎwnfc 2319 class class class wbr 4018 dom cdm 4644 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-dm 4654 |
This theorem is referenced by: dmopab 4856 |
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