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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 2689 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
2 | vex 2689 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 1, 2 | op1sta 5020 | . . . . . . 7 ⊢ ∪ dom {〈𝑦, 𝑧〉} = 𝑦 |
4 | 3 | eleq1i 2205 | . . . . . 6 ⊢ (∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
5 | 4 | biimpri 132 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
6 | 5 | adantr 274 | . . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴) |
7 | 6 | rgen2 2518 | . . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴 |
8 | sneq 3538 | . . . . . . 7 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → {𝑥} = {〈𝑦, 𝑧〉}) | |
9 | 8 | dmeqd 4741 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → dom {𝑥} = dom {〈𝑦, 𝑧〉}) |
10 | 9 | unieqd 3747 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ∪ dom {𝑥} = ∪ dom {〈𝑦, 𝑧〉}) |
11 | 10 | eleq1d 2208 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {〈𝑦, 𝑧〉} ∈ 𝐴)) |
12 | 11 | ralxp 4682 | . . 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 6038 | . . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | |
15 | 14 | reseq1i 4815 | . . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵)) |
16 | ssv 3119 | . . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V | |
17 | resmpt 4867 | . . . . 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 2160 | . . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥}) |
20 | 19 | fmpt 5570 | . 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 1331 ∈ wcel 1480 ∀wral 2416 Vcvv 2686 ⊆ wss 3071 {csn 3527 〈cop 3530 ∪ cuni 3736 ↦ cmpt 3989 × cxp 4537 dom cdm 4539 ↾ cres 4541 ⟶wf 5119 1st c1st 6036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-1st 6038 |
This theorem is referenced by: fo1stresm 6059 1stcof 6061 tx1cn 12438 |
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