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Theorem fo1stres 7947

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 5650	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6841	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1^st ‘⟨𝑥, 𝑦⟩))
4		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op1st 7929	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
7	3, 6	eqtr2di 2783	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f1stres 7945	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9		ffn 6651	. . . . . . . . . . . . 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 7013	. . . . . . . . . . . 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 2831	. . . . . . . . . 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 1931	. . . . . . 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 3935	. . . . 5 ⊢ (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 6658	. . . . . 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 3945	. . . 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 6739	. 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 1541 ∃wex 1780 ∈ wcel 2111 ≠ wne 2928 ⊆ wss 3897 ∅c0 4280 ⟨cop 4579 × cxp 5612 ran crn 5615 ↾ cres 5616 Fn wfn 6476 ⟶wf 6477 –onto→wfo 6479 ‘cfv 6481 1^st c1st 7919

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-1st 7921

This theorem is referenced by: 1stconst 8030 txcmpb 23559

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