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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 5891 | . 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} | |
2 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑥𝑣 | |
3 | dfrnf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 5196 | . . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤 |
6 | nfv 1909 | . . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤 | |
7 | breq1 5152 | . . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤)) | |
8 | 5, 6, 7 | cbvexv1 2332 | . . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤) |
9 | 8 | abbii 2795 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} |
10 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
11 | dfrnf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
12 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
13 | 10, 11, 12 | nfbr 5196 | . . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤 |
14 | 13 | nfex 2312 | . . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤 |
15 | nfv 1909 | . . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦 | |
16 | breq2 5153 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1916 | . . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvabw 2799 | . 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2757 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 {cab 2702 Ⅎwnfc 2875 class class class wbr 5149 ran crn 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-cnv 5686 df-dm 5688 df-rn 5689 |
This theorem is referenced by: rnopab 5956 |
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