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| Mirrors > Home > MPE Home > Th. List > dfdmf | Structured version Visualization version 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 5695 | . 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
| 3 | dfdmf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 4 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝑣 | |
| 5 | 2, 3, 4 | nfbr 5190 | . . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
| 6 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
| 7 | breq2 5147 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
| 8 | 5, 6, 7 | cbvexv1 2344 | . . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦) |
| 9 | 8 | abbii 2809 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} |
| 10 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
| 11 | dfdmf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 12 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 13 | 10, 11, 12 | nfbr 5190 | . . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
| 14 | 13 | nfex 2324 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦 |
| 15 | nfv 1914 | . . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦 | |
| 16 | breq1 5146 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
| 17 | 16 | exbidv 1921 | . . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦)) |
| 18 | 14, 15, 17 | cbvabw 2813 | . 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
| 19 | 1, 9, 18 | 3eqtri 2769 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 {cab 2714 Ⅎwnfc 2890 class class class wbr 5143 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: dmopab 5926 |
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