Theorem f1stres 6393

Description: Mapping of a restriction of the 1^st (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.

Step	Hyp	Ref	Expression
1		vex 2824	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 2824	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op1sta 5269	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑧⟩} = 𝑦
4	3	eleq1i 2304	. . . . . 6 ⊢ (∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
5	4	biimpri 133	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
6	5	adantr 276	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
7	6	rgen2 2636	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8		sneq 3720	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	dmeqd 4983	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
10	9	unieqd 3946	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2307	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
12	11	ralxp 4923	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
13	7, 12	mpbir 146	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴
14		df-1st 6374	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
15	14	reseq1i 5059	. . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3270	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5111	. . . . 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 2259	. . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})
20	19	fmpt 5858	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
21	13, 20	mpbi 145	1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  {csn 3709  ⟨cop 3712  ∪ cuni 3935   ↦ cmpt 4192   × cxp 4772  dom cdm 4774   ↾ cres 4776  ⟶wf 5373  1^st c1st 6372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-1st 6374

This theorem is used by:  fo1stresm  6395  1stcof  6397  tx1cn  15370