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Theorem f1stres 5814
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.)
Assertion
Ref Expression
f1stres (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

Proof of Theorem f1stres
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2577 . . . . . . . 8 𝑦 ∈ V
2 vex 2577 . . . . . . . 8 𝑧 ∈ V
31, 2op1sta 4830 . . . . . . 7 dom {⟨𝑦, 𝑧⟩} = 𝑦
43eleq1i 2119 . . . . . 6 ( dom {⟨𝑦, 𝑧⟩} ∈ 𝐴𝑦𝐴)
54biimpri 128 . . . . 5 (𝑦𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
65adantr 265 . . . 4 ((𝑦𝐴𝑧𝐵) → dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
76rgen2 2422 . . 3 𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8 sneq 3414 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98dmeqd 4565 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
109unieqd 3619 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
1110eleq1d 2122 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( dom {𝑥} ∈ 𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
1211ralxp 4507 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ ∀𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
137, 12mpbir 138 . 2 𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴
14 df-1st 5795 . . . . 5 1st = (𝑥 ∈ V ↦ dom {𝑥})
1514reseq1i 4636 . . . 4 (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 2993 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 4684 . . . . 5 ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥}))
1816, 17ax-mp 7 . . . 4 ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
1915, 18eqtri 2076 . . 3 (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
2019fmpt 5347 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
2113, 20mpbi 137 1 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  wss 2945  {csn 3403  cop 3406   cuni 3608  cmpt 3846   × cxp 4371  dom cdm 4373  cres 4375  wf 4926  1st c1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-1st 5795
This theorem is referenced by:  fo1stresm  5816  1stcof  5818
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