Theorem fo2ndres 7958

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 4281	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		opelxp 5654	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6846	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
4		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op2nd 7940	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
7	3, 6	eqtr2di 2791	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f2ndres 7956	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
9		ffn 6655	. . . . . . . . . . . . 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 7021	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
12	10, 11	mpan 696	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	7, 12	eqeltrd 2839	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	2, 13	sylbir 236	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	14	ex 413	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
16	15	exlimiv 1937	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	1, 16	sylbi 218	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3921	. . . . 5 ⊢ (𝐴 ≠ ∅ → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 6662	. . . . . 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 524	. . . 4 ⊢ (𝐴 ≠ ∅ → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 3930	. . . 4 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 235	. . 3 ⊢ (𝐴 ≠ ∅ → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 8	jctil 524	. 2 ⊢ (𝐴 ≠ ∅ → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 6743	. 2 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 235	1 ⊢ (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934   ⊆ wss 3883  ∅c0 4261  ⟨cop 4561   × cxp 5616  ran crn 5619   ↾ cres 5620   Fn wfn 6480  ⟶wf 6481  –onto→wfo 6483  ‘cfv 6485  2^nd c2nd 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493  df-2nd 7932

This theorem is referenced by:  2ndconst  8040  txcmpb  23627