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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 2755 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 2755 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 5128 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑧〉} = 𝑧 |
4 | 3 | eleq1i 2255 | . . . . . 6 ⊢ (∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 133 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
6 | 5 | adantl 277 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
7 | 6 | rgen2 2576 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 |
8 | sneq 3618 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | rneqd 4871 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ran {𝑥} = ran {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 3835 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2258 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵)) |
12 | 11 | ralxp 4785 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
13 | 7, 12 | mpbir 146 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 6160 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 4918 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 3192 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 4970 | . . . . 5 ⊢ ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥})) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
19 | 15, 18 | eqtri 2210 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 5682 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 145 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 ⊆ wss 3144 {csn 3607 〈cop 3610 ∪ cuni 3824 ↦ cmpt 4079 × cxp 4639 ran crn 4642 ↾ cres 4643 ⟶wf 5227 2nd c2nd 6158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-2nd 6160 |
This theorem is referenced by: fo2ndresm 6181 2ndcof 6183 f2ndf 6245 eucalgcvga 12076 tx2cn 14167 |
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