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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 5233 | . 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} | |
2 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑣 | |
3 | dfrnf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
5 | 2, 3, 4 | nfbr 4629 | . . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤 |
6 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤 | |
7 | breq1 4586 | . . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤)) | |
8 | 5, 6, 7 | cbvex 2260 | . . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤) |
9 | 8 | abbii 2726 | . 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} |
10 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
11 | dfrnf.2 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
12 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
13 | 10, 11, 12 | nfbr 4629 | . . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤 |
14 | 13 | nfex 2140 | . . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤 |
15 | nfv 1830 | . . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦 | |
16 | breq2 4587 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1837 | . . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦)) |
18 | 14, 15, 17 | cbvab 2733 | . 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
19 | 1, 9, 18 | 3eqtri 2636 | 1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∃wex 1695 {cab 2596 Ⅎwnfc 2738 class class class wbr 4583 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: rnopab 5291 |
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