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Mirrors > Home > ILE Home > Th. List > elrnmpt2 | GIF version |
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶) |
elrnmpt2.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elrnmpt2 | ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶) | |
2 | 1 | rnmpt2 5553 | . . 3 ⊢ ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶} |
3 | 2 | eleq2i 2101 | . 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶}) |
4 | elrnmpt2.1 | . . . . . 6 ⊢ 𝐶 ∈ V | |
5 | eleq1 2097 | . . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V)) | |
6 | 4, 5 | mpbiri 157 | . . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V) |
7 | 6 | rexlimivw 2423 | . . . 4 ⊢ (∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V) |
8 | 7 | rexlimivw 2423 | . . 3 ⊢ (∃x ∈ A ∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V) |
9 | eqeq1 2043 | . . . 4 ⊢ (z = 𝐷 → (z = 𝐶 ↔ 𝐷 = 𝐶)) | |
10 | 9 | 2rexbidv 2343 | . . 3 ⊢ (z = 𝐷 → (∃x ∈ A ∃y ∈ B z = 𝐶 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)) |
11 | 8, 10 | elab3 2688 | . 2 ⊢ (𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶} ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶) |
12 | 3, 11 | bitri 173 | 1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 ∃wrex 2301 Vcvv 2551 ran crn 4289 ↦ cmpt2 5457 |
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-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-cnv 4296 df-dm 4298 df-rn 4299 df-oprab 5459 df-mpt2 5460 |
This theorem is referenced by: (None) |
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