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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 5688 | . 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} | |
2 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
3 | dfdmf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 5195 | . . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 5152 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvexv1 2334 | . . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦) |
9 | 8 | abbii 2798 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} |
10 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
11 | dfdmf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
13 | 10, 11, 12 | nfbr 5195 | . . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
14 | 13 | nfex 2313 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦 |
15 | nfv 1910 | . . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦 | |
16 | breq1 5151 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1917 | . . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvabw 2802 | . 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2760 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∃wex 1774 {cab 2705 Ⅎwnfc 2879 class class class wbr 5148 dom cdm 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-dm 5688 |
This theorem is referenced by: dmopab 5918 |
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