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Theorem f1stres 7390
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 3353 . . . . . . . 8 𝑦 ∈ V
2 vex 3353 . . . . . . . 8 𝑧 ∈ V
31, 2op1sta 5802 . . . . . . 7 dom {⟨𝑦, 𝑧⟩} = 𝑦
43eleq1i 2835 . . . . . 6 ( dom {⟨𝑦, 𝑧⟩} ∈ 𝐴𝑦𝐴)
54biimpri 219 . . . . 5 (𝑦𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
65adantr 472 . . . 4 ((𝑦𝐴𝑧𝐵) → dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
76rgen2 3122 . . 3 𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8 sneq 4344 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98dmeqd 5494 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
109unieqd 4604 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
1110eleq1d 2829 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( dom {𝑥} ∈ 𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
1211ralxp 5432 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ ∀𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
137, 12mpbir 222 . 2 𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴
14 df-1st 7366 . . . . 5 1st = (𝑥 ∈ V ↦ dom {𝑥})
1514reseq1i 5561 . . . 4 (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3785 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5626 . . . . 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 2787 . . 3 (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
2019fmpt 6570 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
2113, 20mpbi 221 1 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732  {csn 4334  cop 4340   cuni 4594  cmpt 4888   × cxp 5275  dom cdm 5277  cres 5279  wf 6064  1st c1st 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-1st 7366
This theorem is referenced by:  fo1stres  7392  1stcof  7396  fparlem1  7479  domssex2  8327  domssex  8328  unxpwdom2  8700  1stfcl  17103  tx1cn  21692  xpinpreima  30399  xpinpreima2  30400  1stmbfm  30769  hausgraph  38467
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