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Mirrors > Home > MPE Home > Th. List > dfrnf | Structured version Visualization version GIF version |
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
dfrnf.1 | ⊢ Ⅎ𝑥𝐴 |
dfrnf.2 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
dfrnf | ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrn2 5753 | . 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} | |
2 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑣 | |
3 | dfrnf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 5105 | . . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤 |
6 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤 | |
7 | breq1 5061 | . . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤)) | |
8 | 5, 6, 7 | cbvexv1 2358 | . . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤) |
9 | 8 | abbii 2886 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} |
10 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
11 | dfrnf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
12 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
13 | 10, 11, 12 | nfbr 5105 | . . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤 |
14 | 13 | nfex 2339 | . . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤 |
15 | nfv 1911 | . . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦 | |
16 | breq2 5062 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1918 | . . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvabw 2890 | . 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2848 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1776 {cab 2799 Ⅎwnfc 2961 class class class wbr 5058 ran crn 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: rnopab 5820 |
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