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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 5285 | . 2 ⊢ (∀x ∈ A B ∈ 𝑉 → (x ∈ A ↦ B) = ∪ x ∈ A {〈x, B〉}) | |
2 | eqid 2037 | . . . . 5 ⊢ (x ∈ A ↦ 〈x, B〉) = (x ∈ A ↦ 〈x, B〉) | |
3 | 2 | rnmpt 4525 | . . . 4 ⊢ ran (x ∈ A ↦ 〈x, B〉) = {y ∣ ∃x ∈ A y = 〈x, B〉} |
4 | elsn 3382 | . . . . . 6 ⊢ (y ∈ {〈x, B〉} ↔ y = 〈x, B〉) | |
5 | 4 | rexbii 2325 | . . . . 5 ⊢ (∃x ∈ A y ∈ {〈x, B〉} ↔ ∃x ∈ A y = 〈x, B〉) |
6 | 5 | abbii 2150 | . . . 4 ⊢ {y ∣ ∃x ∈ A y ∈ {〈x, B〉}} = {y ∣ ∃x ∈ A y = 〈x, B〉} |
7 | 3, 6 | eqtr4i 2060 | . . 3 ⊢ ran (x ∈ A ↦ 〈x, B〉) = {y ∣ ∃x ∈ A y ∈ {〈x, B〉}} |
8 | df-iun 3650 | . . 3 ⊢ ∪ x ∈ A {〈x, B〉} = {y ∣ ∃x ∈ A y ∈ {〈x, B〉}} | |
9 | 7, 8 | eqtr4i 2060 | . 2 ⊢ ran (x ∈ A ↦ 〈x, B〉) = ∪ x ∈ A {〈x, B〉} |
10 | 1, 9 | syl6eqr 2087 | 1 ⊢ (∀x ∈ A B ∈ 𝑉 → (x ∈ A ↦ B) = ran (x ∈ A ↦ 〈x, B〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 {cab 2023 ∀wral 2300 ∃wrex 2301 {csn 3367 〈cop 3370 ∪ ciun 3648 ↦ cmpt 3809 ran crn 4289 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 |
This theorem is referenced by: resfunexg 5325 |
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