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Theorem f1stres 7723
 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 3413 . . . . . . . 8 𝑦 ∈ V
2 vex 3413 . . . . . . . 8 𝑧 ∈ V
31, 2op1sta 6059 . . . . . . 7 dom {⟨𝑦, 𝑧⟩} = 𝑦
43eleq1i 2842 . . . . . 6 ( dom {⟨𝑦, 𝑧⟩} ∈ 𝐴𝑦𝐴)
54biimpri 231 . . . . 5 (𝑦𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
65adantr 484 . . . 4 ((𝑦𝐴𝑧𝐵) → dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
76rgen2 3132 . . 3 𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8 sneq 4535 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98dmeqd 5751 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
109unieqd 4815 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
1110eleq1d 2836 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( dom {𝑥} ∈ 𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
1211ralxp 5687 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ ∀𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
137, 12mpbir 234 . 2 𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴
14 df-1st 7699 . . . . 5 1st = (𝑥 ∈ V ↦ dom {𝑥})
1514reseq1i 5824 . . . 4 (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3918 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5882 . . . . 5 ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥}))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
1915, 18eqtri 2781 . . 3 (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
2019fmpt 6871 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
2113, 20mpbi 233 1 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ⊆ wss 3860  {csn 4525  ⟨cop 4531  ∪ cuni 4801   ↦ cmpt 5116   × cxp 5526  dom cdm 5528   ↾ cres 5530  ⟶wf 6336  1st c1st 7697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-1st 7699 This theorem is referenced by:  fo1stres  7725  1stcof  7729  fparlem1  7818  domssex2  8712  domssex  8713  unxpwdom2  9098  1stfcl  17526  tx1cn  22322  xpinpreima  31389  xpinpreima2  31390  1stmbfm  31758  hausgraph  40564
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