Theorem fo2ndres 7974

Description: Onto mapping of a restriction of the 2^nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
fo2ndres	⊢ (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Proof of Theorem fo2ndres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4312	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		opelxp 5667	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6859	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
4		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op2nd 7956	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
7	3, 6	eqtr2di 2781	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f2ndres 7972	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
9		ffn 6670	. . . . . . . . . . . . 13 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
11		fnfvelrn 7034	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
12	10, 11	mpan 690	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	7, 12	eqeltrd 2828	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	2, 13	sylbir 235	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	14	ex 412	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
16	15	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	1, 16	sylbi 217	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3949	. . . . 5 ⊢ (𝐴 ≠ ∅ → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 6677	. . . . . 6 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
20	8, 19	ax-mp 5	. . . . 5 ⊢ ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
21	18, 20	jctil 519	. . . 4 ⊢ (𝐴 ≠ ∅ → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 3959	. . . 4 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 234	. . 3 ⊢ (𝐴 ≠ ∅ → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 8	jctil 519	. 2 ⊢ (𝐴 ≠ ∅ → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 6758	. 2 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 234	1 ⊢ (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925   ⊆ wss 3911  ∅c0 4292  ⟨cop 4591   × cxp 5629  ran crn 5632   ↾ cres 5633   Fn wfn 6494  ⟶wf 6495  –onto→wfo 6497  ‘cfv 6499  2^nd c2nd 7946

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-2nd 7948

This theorem is referenced by:  2ndconst  8057  txcmpb  23507