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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 5276 | . 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} | |
2 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
3 | dfdmf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 4851 | . . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1992 | . . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 4808 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvex 2417 | . . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦) |
9 | 8 | abbii 2877 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} |
10 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
11 | dfdmf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
13 | 10, 11, 12 | nfbr 4851 | . . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
14 | 13 | nfex 2301 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦 |
15 | nfv 1992 | . . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦 | |
16 | breq1 4807 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1999 | . . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvab 2884 | . 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2786 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∃wex 1853 {cab 2746 Ⅎwnfc 2889 class class class wbr 4804 dom cdm 5266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-dm 5276 |
This theorem is referenced by: dmopab 5490 |
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