Theorem f1stres 6142

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 2734	. . . . . . . 8 ⊢ 𝑦 ∈ V
2		vex 2734	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	1, 2	op1sta 5094	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑧⟩} = 𝑦
4	3	eleq1i 2237	. . . . . 6 ⊢ (∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
5	4	biimpri 132	. . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
6	5	adantr 274	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
7	6	rgen2 2557	. . 3 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴
8		sneq 3595	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
9	8	dmeqd 4814	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → dom {𝑥} = dom {⟨𝑦, 𝑧⟩})
10	9	unieqd 3808	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑧⟩})
11	10	eleq1d 2240	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∪ dom {𝑥} ∈ 𝐴 ↔ ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴))
12	11	ralxp 4755	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∪ dom {⟨𝑦, 𝑧⟩} ∈ 𝐴)
13	7, 12	mpbir 145	. 2 ⊢ ∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴
14		df-1st 6123	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
15	14	reseq1i 4888	. . . 4 ⊢ (1st ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ∪ dom {𝑥}) ↾ (𝐴 × 𝐵))
16		ssv 3170	. . . . 5 ⊢ (𝐴 × 𝐵) ⊆ V
17		resmpt 4940	. . . . 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 2192	. . 3 ⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ dom {𝑥})
20	19	fmpt 5650	. 2 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)∪ dom {𝑥} ∈ 𝐴 ↔ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴)
21	13, 20	mpbi 144	1 ⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴

Syntax hints:   = wceq 1349   ∈ wcel 2142  ∀wral 2449  Vcvv 2731   ⊆ wss 3122  {csn 3584  ⟨cop 3587  ∪ cuni 3797   ↦ cmpt 4051   × cxp 4610  dom cdm 4612   ↾ cres 4614  ⟶wf 5196  1^st c1st 6121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-fv 5208  df-1st 6123

This theorem is referenced by:  fo1stresm  6144  1stcof  6146  tx1cn  13148