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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 2622 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 2622 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 4915 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑧〉} = 𝑧 |
4 | 3 | eleq1i 2153 | . . . . . 6 ⊢ (∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 131 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
6 | 5 | adantl 271 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
7 | 6 | rgen2 2459 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 |
8 | sneq 3457 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | rneqd 4664 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ran {𝑥} = ran {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 3664 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2156 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵)) |
12 | 11 | ralxp 4579 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
13 | 7, 12 | mpbir 144 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 5912 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 4709 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 3046 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 4760 | . . . . 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 2108 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 5449 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 143 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∈ wcel 1438 ∀wral 2359 Vcvv 2619 ⊆ wss 2999 {csn 3446 〈cop 3449 ∪ cuni 3653 ↦ cmpt 3899 × cxp 4436 ran crn 4439 ↾ cres 4440 ⟶wf 5011 2nd c2nd 5910 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-2nd 5912 |
This theorem is referenced by: fo2ndresm 5933 2ndcof 5935 f2ndf 5991 eucialgcvga 11318 |
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