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Mirrors > Home > ILE Home > Th. List > rnmpt | GIF version |
Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
rnmpt | ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnopab 4781 | . 2 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | df-mpt 3986 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
4 | 2, 3 | eqtri 2158 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
5 | 4 | rneqi 4762 | . 2 ⊢ ran 𝐹 = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
6 | df-rex 2420 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
7 | 6 | abbii 2253 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
8 | 1, 5, 7 | 3eqtr4i 2168 | 1 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2123 ∃wrex 2415 {copab 3983 ↦ cmpt 3984 ran crn 4535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-mpt 3986 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: elrnmpt 4783 elrnmpt1 4785 elrnmptg 4786 dfiun3g 4791 dfiin3g 4792 fnrnfv 5461 fmpt 5563 fnasrn 5591 fnasrng 5593 fliftf 5693 abrexex 6008 abrexexg 6009 fo1st 6048 fo2nd 6049 qliftf 6507 negfi 10992 restco 12332 |
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