Theorem f1stres 6383

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 5264	. . . . . . 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 3716	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	dmeqd 4978	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
10	9	unieqd 3941	. . . . 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 4918	. . 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 6364	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
15	14	reseq1i 5054	. . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3270	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 5106	. . . . 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 5849	. 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

Syntax hints:   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  {csn 3705  ⟨cop 3708  ∪ cuni 3930   ↦ cmpt 4187   × cxp 4767  dom cdm 4769   ↾ cres 4771  ⟶wf 5368  1^st c1st 6362

This theorem was proved from 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 4244  ax-pow 4306  ax-pr 4341

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-1st 6364

This theorem is referenced by:  fo1stresm  6385  1stcof  6387  tx1cn  15293