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Mirrors > Home > ILE Home > Th. List > f1stres | 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 2644 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 2644 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op1sta 4956 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑧〉} = 𝑦 |
4 | 3 | eleq1i 2165 | . . . . . 6 ⊢ (∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
5 | 4 | biimpri 132 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
6 | 5 | adantr 272 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
7 | 6 | rgen2 2477 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 |
8 | sneq 3485 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | dmeqd 4679 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → dom {𝑥} = dom {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 3694 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2168 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴)) |
12 | 11 | ralxp 4620 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
13 | 7, 12 | mpbir 145 | . 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 |
14 | df-1st 5969 | . . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
15 | 14 | reseq1i 4751 | . . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 3069 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 4803 | . . . . 5 ⊢ ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})) | |
18 | 16, 17 | ax-mp 7 | . . . 4 ⊢ ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥}) |
19 | 15, 18 | eqtri 2120 | . . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥}) |
20 | 19 | fmpt 5502 | . 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴) |
21 | 13, 20 | mpbi 144 | 1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∈ wcel 1448 ∀wral 2375 Vcvv 2641 ⊆ wss 3021 {csn 3474 〈cop 3477 ∪ cuni 3683 ↦ cmpt 3929 × cxp 4475 dom cdm 4477 ↾ cres 4479 ⟶wf 5055 1st c1st 5967 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-1st 5969 |
This theorem is referenced by: fo1stresm 5990 1stcof 5992 tx1cn 12219 |
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