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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 2554 | . . . . . . . 8 ⊢ y ∈ V | |
2 | vex 2554 | . . . . . . . 8 ⊢ z ∈ V | |
3 | 1, 2 | op2nda 4748 | . . . . . . 7 ⊢ ∪ ran {〈y, z〉} = z |
4 | 3 | eleq1i 2100 | . . . . . 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 2399 | . . 3 ⊢ ∀y ∈ A ∀z ∈ B ∪ ran {〈y, z〉} ∈ B |
8 | sneq 3378 | . . . . . . 7 ⊢ (x = 〈y, z〉 → {x} = {〈y, z〉}) | |
9 | 8 | rneqd 4506 | . . . . . 6 ⊢ (x = 〈y, z〉 → ran {x} = ran {〈y, z〉}) |
10 | 9 | unieqd 3582 | . . . . 5 ⊢ (x = 〈y, z〉 → ∪ ran {x} = ∪ ran {〈y, z〉}) |
11 | 10 | eleq1d 2103 | . . . 4 ⊢ (x = 〈y, z〉 → (∪ ran {x} ∈ B ↔ ∪ ran {〈y, z〉} ∈ B)) |
12 | 11 | ralxp 4422 | . . 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 5710 | . . . . 5 ⊢ 2nd = (x ∈ V ↦ ∪ ran {x}) | |
15 | 14 | reseq1i 4551 | . . . 4 ⊢ (2nd ↾ (A × B)) = ((x ∈ V ↦ ∪ ran {x}) ↾ (A × B)) |
16 | ssv 2959 | . . . . 5 ⊢ (A × B) ⊆ V | |
17 | resmpt 4599 | . . . . 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 2057 | . . 3 ⊢ (2nd ↾ (A × B)) = (x ∈ (A × B) ↦ ∪ ran {x}) |
20 | 19 | fmpt 5262 | . 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 1242 ∈ wcel 1390 ∀wral 2300 Vcvv 2551 ⊆ wss 2911 {csn 3367 〈cop 3370 ∪ cuni 3571 ↦ cmpt 3809 × cxp 4286 ran crn 4289 ↾ cres 4290 ⟶wf 4841 2nd c2nd 5708 |
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-rab 2309 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-uni 3572 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-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-2nd 5710 |
This theorem is referenced by: fo2ndresm 5731 2ndcof 5733 f2ndf 5789 |
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