Theorem fo2ndresm 5789

Description: Onto mapping of a restriction of the 2^nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Assertion

Ref	Expression
fo2ndresm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fo2ndresm

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2100	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1795	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		opelxp 4374	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5198	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5		vex 2560	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2560	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op2nd 5774	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
8	4, 7	syl6req 2089	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f2ndres 5787	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10		ffn 5046	. . . . . . . . . . . . 13 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
11	9, 10	ax-mp 7	. . . . . . . . . . . 12 ⊢ (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12		fnfvelrn 5299	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	11, 12	mpan 400	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2114	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 125	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
16	15	ex 108	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1489	. . . . . . 7 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 2951	. . . . . 6 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 5052	. . . . . . 7 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
20	9, 19	ax-mp 7	. . . . . 6 ⊢ ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
21	18, 20	jctil 295	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 2960	. . . . 5 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 137	. . . 4 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 9	jctil 295	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 5110	. . 3 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 137	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
27	2, 26	sylbir 125	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ⊆ wss 2917  ⟨cop 3378   × cxp 4343  ran crn 4346   ↾ cres 4347   Fn wfn 4897  ⟶wf 4898  –onto→wfo 4900  ‘cfv 4902  2^nd c2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-2nd 5768

This theorem is referenced by:  2ndconst  5843