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Mirrors > Home > ILE Home > Th. List > fnasrng | GIF version |
Description: A function expressed as the range of another function. (Contributed by Jim Kingdon, 9-Jan-2019.) |
Ref | Expression |
---|---|
fnasrng | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmptg 5363 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) | |
2 | eqid 2081 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) | |
3 | 2 | rnmpt 4600 | . . . 4 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
4 | velsn 3415 | . . . . . 6 ⊢ (𝑦 ∈ {〈𝑥, 𝐵〉} ↔ 𝑦 = 〈𝑥, 𝐵〉) | |
5 | 4 | rexbii 2373 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉) |
6 | 5 | abbii 2194 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 〈𝑥, 𝐵〉} |
7 | 3, 6 | eqtr4i 2104 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} |
8 | df-iun 3680 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ {〈𝑥, 𝐵〉}} | |
9 | 7, 8 | eqtr4i 2104 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
10 | 1, 9 | syl6eqr 2131 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑥 ∈ 𝐴 ↦ 〈𝑥, 𝐵〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 {csn 3398 〈cop 3401 ∪ ciun 3678 ↦ cmpt 3839 ran crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 |
This theorem is referenced by: resfunexg 5403 |
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