fo1stres - Metamath Proof Explorer

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Theorem fo1stres 8018

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 4300	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		opelxp 5691	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6900	. . . . . . . . . . . 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 8000	. . . . . . . . . . . 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 8016	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9		ffn 6705	. . . . . . . . . . . . 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 7076	. . . . . . . . . . . 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 1963	. . . . . . 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 3937	. . . . 5 ⊢ (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 6713	. . . . . 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 3946	. . . 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 6796	. 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 1570 ∃wex 1812 ∈ wcel 2145 ≠ wne 2955 ⊆ wss 3899 ∅c0 4279 ⟨cop 4590 × cxp 5653 ran crn 5656 ↾ cres 5657 Fn wfn 6530 ⟶wf 6531 –onto→wfo 6533 ‘cfv 6535 1^st c1st 7990

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-fo 6541 df-fv 6543 df-1st 7992

This theorem is used by: 1stconst 8102 txcmpb 23902

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