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Mirrors > Home > ILE Home > Th. List > fnasrn | GIF version |
Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfmpt.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fnasrn | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmpt.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | dfmpt 5605 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
3 | eqid 2140 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | |
4 | 3 | rnmpt 4795 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
5 | velsn 3549 | . . . . . 6 ⊢ (𝑦 ∈ {〈𝑥, 𝐵〉} ↔ 𝑦 = 〈𝑥, 𝐵〉) | |
6 | 5 | rexbii 2445 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉) |
7 | 6 | abbii 2256 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
8 | 4, 7 | eqtr4i 2164 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} |
9 | df-iun 3823 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | |
10 | 8, 9 | eqtr4i 2164 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
11 | 2, 10 | eqtr4i 2164 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 {cab 2126 ∃wrex 2418 Vcvv 2689 {csn 3532 〈cop 3535 ∪ ciun 3821 ↦ cmpt 3997 ran crn 4548 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 |
This theorem is referenced by: idref 5666 |
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