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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 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 3176 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op2nda 5538 | . . . . . . 7 ⊢ ∪ ran {〈𝑦, 𝑧〉} = 𝑧 |
4 | 3 | eleq1i 2679 | . . . . . 6 ⊢ (∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 ↔ 𝑧 ∈ 𝐵) |
5 | 4 | biimpri 217 | . . . . 5 ⊢ (𝑧 ∈ 𝐵 → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
7 | 6 | rgen2 2958 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵 |
8 | sneq 4135 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | rneqd 5274 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ran {𝑥} = ran {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 4382 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ ran {𝑥} = ∪ ran {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ ran {𝑥} ∈ 𝐵 ↔ ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵)) |
12 | 11 | ralxp 5185 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ ran {〈𝑦, 𝑧〉} ∈ 𝐵) |
13 | 7, 12 | mpbir 220 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 |
14 | df-2nd 7060 | . . . . 5 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
15 | 14 | reseq1i 5313 | . . . 4 ⊢ (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ ran {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 3588 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 5369 | . . . . 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 2632 | . . 3 ⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ran {𝑥}) |
20 | 19 | fmpt 6289 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵) |
21 | 13, 20 | mpbi 219 | 1 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 {csn 4125 〈cop 4131 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ↾ cres 5040 ⟶wf 5800 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-2nd 7060 |
This theorem is referenced by: fo2ndres 7084 2ndcof 7088 fparlem2 7165 f2ndf 7170 eucalgcvga 15137 2ndfcl 16661 gaid 17555 tx2cn 21223 txkgen 21265 xpinpreima 29280 xpinpreima2 29281 2ndmbfm 29650 filnetlem4 31546 hausgraph 36809 |
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