Theorem fo2ndresm 6141

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 2233	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1911	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		opelxp 4641	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5520	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op2nd 6126	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
8	4, 7	eqtr2di 2220	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f2ndres 6139	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10		ffn 5347	. . . . . . . . . . . . 13 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12		fnfvelrn 5628	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	11, 12	mpan 422	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2247	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 134	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
16	15	ex 114	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1591	. . . . . . 7 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3153	. . . . . 6 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 5356	. . . . . . 7 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
20	9, 19	ax-mp 5	. . . . . 6 ⊢ ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
21	18, 20	jctil 310	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 3162	. . . . 5 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 133	. . . 4 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 9	jctil 310	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 5424	. . 3 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 133	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
27	2, 26	sylbir 134	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ⊆ wss 3121  ⟨cop 3586   × cxp 4609  ran crn 4612   ↾ cres 4613   Fn wfn 5193  ⟶wf 5194  –onto→wfo 5196  ‘cfv 5198  2^nd c2nd 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-2nd 6120

This theorem is referenced by:  2ndconst  6201