Theorem f1stres 7081

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 3176	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op1sta 5535	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑧⟩} = 𝑦
4	3	eleq1i 2679	. . . . . 6 ⊢ (∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
5	4	biimpri 217	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
6	5	adantr 480	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
7	6	rgen2 2958	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8		sneq 4135	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	dmeqd 5248	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
10	9	unieqd 4382	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2672	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
12	11	ralxp 5185	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
13	7, 12	mpbir 220	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴
14		df-1st 7059	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
15	14	reseq1i 5313	. . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3588	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5369	. . . . 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 2632	. . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})
20	19	fmpt 6289	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
21	13, 20	mpbi 219	1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ↾ cres 5040  ⟶wf 5800  1^st c1st 7057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059

This theorem is referenced by:  fo1stres  7083  1stcof  7087  fparlem1  7164  domssex2  8005  domssex  8006  unxpwdom2  8376  1stfcl  16660  tx1cn  21222  xpinpreima  29280  xpinpreima2  29281  1stmbfm  29649  hausgraph  36809