fo1stres - Metamath Proof Explorer

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Theorem fo1stres 8027

Description: Onto mapping of a restriction of the 1^st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

**Assertion**
Ref	Expression
fo1stres	⊢ (𝐵 ≠ ∅ → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Proof of Theorem fo1stres Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4299	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		opelxp 5686	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6904	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1^st ‘⟨𝑥, 𝑦⟩))
4		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op1st 8009	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
7	3, 6	eqtr2di 2812	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f1stres 8025	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9		ffn 6709	. . . . . . . . . . . . 13 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
11		fnfvelrn 7080	. . . . . . . . . . . 12 ⊢ (((1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
12	10, 11	mpan 703	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
13	7, 12	eqeltrd 2860	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	2, 13	sylbir 238	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	14	expcom 419	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
16	15	exlimiv 1962	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	1, 16	sylbi 220	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3914	. . . . 5 ⊢ (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 6717	. . . . . 6 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
20	8, 19	ax-mp 5	. . . . 5 ⊢ ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
21	18, 20	jctil 529	. . . 4 ⊢ (𝐵 ≠ ∅ → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3923	. . . 4 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 237	. . 3 ⊢ (𝐵 ≠ ∅ → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 8	jctil 529	. 2 ⊢ (𝐵 ≠ ∅ → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 6800	. 2 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴 ↔ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
26	24, 25	sylibr 237	1 ⊢ (𝐵 ≠ ∅ → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ≠ wne 2955 ⊆ wss 3895 ∅c0 4278 ⟨cop 4589 × cxp 5648 ran crn 5651 ↾ cres 5652 Fn wfn 6533 ⟶wf 6534 –onto→wfo 6536 ‘cfv 6538 1^st c1st 7999

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fo 6544 df-fv 6546 df-1st 8001

This theorem is used by: 1stconst 8111 txcmpb 23963

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