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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 5673 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
3 | eqid 2170 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | |
4 | 3 | rnmpt 4859 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
5 | velsn 3600 | . . . . . 6 ⊢ (𝑦 ∈ {〈𝑥, 𝐵〉} ↔ 𝑦 = 〈𝑥, 𝐵〉) | |
6 | 5 | rexbii 2477 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉) |
7 | 6 | abbii 2286 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
8 | 4, 7 | eqtr4i 2194 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} |
9 | df-iun 3875 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | |
10 | 8, 9 | eqtr4i 2194 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
11 | 2, 10 | eqtr4i 2194 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 {cab 2156 ∃wrex 2449 Vcvv 2730 {csn 3583 〈cop 3586 ∪ ciun 3873 ↦ cmpt 4050 ran crn 4612 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 |
This theorem is referenced by: idref 5736 |
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