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Theorem fo1stres 8014

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 4328	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		opelxp 5690	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6895	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1^st ‘⟨𝑥, 𝑦⟩))
4		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3463	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op1st 7996	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
7	3, 6	eqtr2di 2787	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f1stres 8012	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9		ffn 6706	. . . . . . . . . . . . 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 7070	. . . . . . . . . . . 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 690	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
13	7, 12	eqeltrd 2834	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	2, 13	sylbir 235	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	14	expcom 413	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
16	15	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	1, 16	sylbi 217	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3964	. . . . 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 519	. . . 4 ⊢ (𝐵 ≠ ∅ → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3974	. . . 4 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 234	. . 3 ⊢ (𝐵 ≠ ∅ → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 8	jctil 519	. 2 ⊢ (𝐵 ≠ ∅ → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 6794	. 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 234	1 ⊢ (𝐵 ≠ ∅ → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ⊆ wss 3926 ∅c0 4308 ⟨cop 4607 × cxp 5652 ran crn 5655 ↾ cres 5656 Fn wfn 6526 ⟶wf 6527 –onto→wfo 6529 ‘cfv 6531 1^st c1st 7986

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fo 6537 df-fv 6539 df-1st 7988

This theorem is referenced by: 1stconst 8099 txcmpb 23582

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