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Mirrors > Home > MPE Home > Th. List > f2ndres | Structured version Visualization version 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 3478 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 3478 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 6227 | . . . . . . 7 ⊢ ∪ ran {⟨𝑦, 𝑧⟩} = 𝑧 |
4 | 3 | eleq1i 2824 | . . . . . 6 ⊢ (∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 227 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵) |
6 | 5 | adantl 482 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵) |
7 | 6 | rgen2 3197 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵 |
8 | sneq 4638 | . . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩}) | |
9 | 8 | rneqd 5937 | . . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩}) |
10 | 9 | unieqd 4922 | . . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ ran {𝑥} = ∪ ran {⟨𝑦, 𝑧⟩}) |
11 | 10 | eleq1d 2818 | . . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)) |
12 | 11 | ralxp 5841 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {⟨𝑦, 𝑧⟩} ∈ 𝐵) |
13 | 7, 12 | mpbir 230 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 7978 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 5977 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 4006 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 6037 | . . . . 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 2760 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 7111 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 229 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3948 {csn 4628 ⟨cop 4634 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5674 ran crn 5677 ↾ cres 5678 ⟶wf 6539 2nd c2nd 7976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 df-2nd 7978 |
This theorem is referenced by: fo2ndres 8004 2ndcof 8008 fparlem2 8101 f2ndf 8108 eucalgcvga 16527 2ndfcl 18154 gaid 19204 tx2cn 23334 txkgen 23376 xpinpreima 33172 xpinpreima2 33173 2ndmbfm 33546 filnetlem4 35569 hausgraph 42256 |
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