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Mirrors > Home > MPE Home > Th. List > f1stres | Structured version Visualization version GIF version |
Description: Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
f1stres | ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 3481 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op1sta 6246 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑧〉} = 𝑦 |
4 | 3 | eleq1i 2829 | . . . . . 6 ⊢ (∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
5 | 4 | biimpri 228 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
7 | 6 | rgen2 3196 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 |
8 | sneq 4640 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | dmeqd 5918 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → dom {𝑥} = dom {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 4924 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2823 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴)) |
12 | 11 | ralxp 5854 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
13 | 7, 12 | mpbir 231 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 |
14 | df-1st 8012 | . . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
15 | 14 | reseq1i 5995 | . . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 4019 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 6056 | . . . . 5 ⊢ ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥}) |
19 | 15, 18 | eqtri 2762 | . . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥}) |
20 | 19 | fmpt 7129 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴) |
21 | 13, 20 | mpbi 230 | 1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⊆ wss 3962 {csn 4630 〈cop 4636 ∪ cuni 4911 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 ↾ cres 5690 ⟶wf 6558 1st c1st 8010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 df-fn 6565 df-f 6566 df-1st 8012 |
This theorem is referenced by: fo1stres 8038 1stcof 8042 fparlem1 8135 domssex2 9175 domssex 9176 unxpwdom2 9625 1stfcl 18252 tx1cn 23632 xpinpreima 33866 xpinpreima2 33867 1stmbfm 34241 hausgraph 43193 |
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