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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 5644 | . 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
3 | dfdmf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
4 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑣 | |
5 | 2, 3, 4 | nfbr 5153 | . . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣 |
6 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦 | |
7 | breq2 5110 | . . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦)) | |
8 | 5, 6, 7 | cbvexv1 2339 | . . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦) |
9 | 8 | abbii 2803 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} |
10 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
11 | dfdmf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
13 | 10, 11, 12 | nfbr 5153 | . . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦 |
14 | 13 | nfex 2318 | . . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦 |
15 | nfv 1918 | . . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦 | |
16 | breq1 5109 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1925 | . . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvabw 2807 | . 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2765 | 1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∃wex 1782 {cab 2710 Ⅎwnfc 2884 class class class wbr 5106 dom cdm 5634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-dm 5644 |
This theorem is referenced by: dmopab 5872 |
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