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Theorem f1stres 7357
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 3343 . . . . . . . 8 𝑦 ∈ V
2 vex 3343 . . . . . . . 8 𝑧 ∈ V
31, 2op1sta 5777 . . . . . . 7 dom {⟨𝑦, 𝑧⟩} = 𝑦
43eleq1i 2830 . . . . . 6 ( dom {⟨𝑦, 𝑧⟩} ∈ 𝐴𝑦𝐴)
54biimpri 218 . . . . 5 (𝑦𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
65adantr 472 . . . 4 ((𝑦𝐴𝑧𝐵) → dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
76rgen2 3113 . . 3 𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8 sneq 4331 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98dmeqd 5481 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
109unieqd 4598 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
1110eleq1d 2824 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( dom {𝑥} ∈ 𝐴 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
1211ralxp 5419 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ ∀𝑦𝐴𝑧𝐵 dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
137, 12mpbir 221 . 2 𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴
14 df-1st 7333 . . . . 5 1st = (𝑥 ∈ V ↦ dom {𝑥})
1514reseq1i 5547 . . . 4 (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ dom {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3766 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5607 . . . . 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 2782 . . 3 (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ dom {𝑥})
2019fmpt 6544 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
2113, 20mpbi 220 1 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  {csn 4321  cop 4327   cuni 4588  cmpt 4881   × cxp 5264  dom cdm 5266  cres 5268  wf 6045  1st c1st 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-1st 7333
This theorem is referenced by:  fo1stres  7359  1stcof  7363  fparlem1  7445  domssex2  8285  domssex  8286  unxpwdom2  8658  1stfcl  17038  tx1cn  21614  xpinpreima  30261  xpinpreima2  30262  1stmbfm  30631  hausgraph  38292
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