Theorem fo2ndresm 6386

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 2301	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	cbvexv 1974	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
3		opelxp 4799	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5714	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op2nd 6371	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
8	4, 7	eqtr2di 2288	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f2ndres 6384	. . . . . . . . . . . . 13 ⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10		ffn 5528	. . . . . . . . . . . . 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 5831	. . . . . . . . . . . 12 ⊢ (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
13	11, 12	mpan 428	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2315	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 135	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
16	15	ex 115	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (𝑣 ∈ 𝐵 → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3254	. . . . . 6 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19		frn 5537	. . . . . . 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 312	. . . . 5 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22		eqss 3263	. . . . 5 ⊢ (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 134	. . . 4 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
24	23, 9	jctil 312	. . 3 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25		dffo2 5614	. . 3 ⊢ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
26	24, 25	sylibr 134	. 2 ⊢ (∃𝑢 𝑢 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)
27	2, 26	sylbir 135	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ⊆ wss 3220  ⟨cop 3708   × cxp 4767  ran crn 4770   ↾ cres 4771   Fn wfn 5367  ⟶wf 5368  –onto→wfo 5370  ‘cfv 5372  2^nd c2nd 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-2nd 6365

This theorem is referenced by:  2ndconst  6448