Theorem f1stres 7981

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 3477	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 3477	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op1sta 6213	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑧⟩} = 𝑦
4	3	eleq1i 2823	. . . . . 6 ⊢ (∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
5	4	biimpri 227	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
6	5	adantr 481	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
7	6	rgen2 3196	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8		sneq 4632	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	dmeqd 5897	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
10	9	unieqd 4915	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2817	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
12	11	ralxp 5833	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
13	7, 12	mpbir 230	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴
14		df-1st 7957	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
15	14	reseq1i 5969	. . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 4002	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 6027	. . . . 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 2759	. . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})
20	19	fmpt 7094	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
21	13, 20	mpbi 229	1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3473   ⊆ wss 3944  {csn 4622  ⟨cop 4628  ∪ cuni 4901   ↦ cmpt 5224   × cxp 5667  dom cdm 5669   ↾ cres 5671  ⟶wf 6528  1^st c1st 7955

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6534  df-fn 6535  df-f 6536  df-1st 7957

This theorem is referenced by:  fo1stres  7983  1stcof  7987  fparlem1  8080  domssex2  9120  domssex  9121  unxpwdom2  9565  1stfcl  18131  tx1cn  23042  xpinpreima  32717  xpinpreima2  32718  1stmbfm  33090  hausgraph  41725