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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 5759 | . 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 5113 | . . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤 |
6 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤 | |
7 | breq1 5069 | . . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤)) | |
8 | 5, 6, 7 | cbvexv1 2362 | . . 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 5113 | . . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤 |
14 | 13 | nfex 2343 | . . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤 |
15 | nfv 1915 | . . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦 | |
16 | breq2 5070 | . . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦)) | |
17 | 16 | exbidv 1922 | . . 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 1537 ∃wex 1780 {cab 2799 Ⅎwnfc 2961 class class class wbr 5066 ran crn 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: rnopab 5826 |
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