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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 3482 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 3482 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 6250 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑧〉} = 𝑧 |
4 | 3 | eleq1i 2830 | . . . . . 6 ⊢ (∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 228 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
7 | 6 | rgen2 3197 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 |
8 | sneq 4641 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | rneqd 5952 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ran {𝑥} = ran {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 4925 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2824 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵)) |
12 | 11 | ralxp 5855 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
13 | 7, 12 | mpbir 231 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 8014 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 5996 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 4020 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 6057 | . . . . 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 2763 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 7130 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 230 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 {csn 4631 〈cop 4637 ∪ cuni 4912 ↦ cmpt 5231 × cxp 5687 ran crn 5690 ↾ cres 5691 ⟶wf 6559 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-2nd 8014 |
This theorem is referenced by: fo2ndres 8040 2ndcof 8044 fparlem2 8137 f2ndf 8144 eucalgcvga 16620 2ndfcl 18254 gaid 19330 tx2cn 23634 txkgen 23676 xpinpreima 33867 xpinpreima2 33868 2ndmbfm 34243 filnetlem4 36364 hausgraph 43194 |
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