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Mirrors > Home > ILE Home > Th. List > f2ndres | GIF version |
Description: Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
f2ndres | ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 2560 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 4805 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑧〉} = 𝑧 |
4 | 3 | eleq1i 2103 | . . . . . 6 ⊢ (∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 124 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
6 | 5 | adantl 262 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
7 | 6 | rgen2 2405 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 |
8 | sneq 3386 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | rneqd 4563 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ran {𝑥} = ran {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 3591 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵)) |
12 | 11 | ralxp 4479 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
13 | 7, 12 | mpbir 134 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 5768 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 4608 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 2965 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 4656 | . . . . 5 ⊢ ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})) | |
18 | 16, 17 | ax-mp 7 | . . . 4 ⊢ ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
19 | 15, 18 | eqtri 2060 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 5319 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 133 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 ∀wral 2306 Vcvv 2557 ⊆ wss 2917 {csn 3375 〈cop 3378 ∪ cuni 3580 ↦ cmpt 3818 × cxp 4343 ran crn 4346 ↾ cres 4347 ⟶wf 4898 2nd c2nd 5766 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-2nd 5768 |
This theorem is referenced by: fo2ndresm 5789 2ndcof 5791 f2ndf 5847 |
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