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Theorem fo1stresm 5788

Description: Onto mapping of a restriction of the 1^st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

**Assertion**
Ref	Expression
fo1stresm	⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Distinct variable group: 𝑦,𝐵 Allowed substitution hint: 𝐴(𝑦)

Proof of Theorem fo1stresm Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2100	. . 3 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
2	1	cbvexv 1795	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
3		opelxp 4374	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5198	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1^st ‘⟨𝑢, 𝑣⟩))
5		vex 2560	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2560	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op1st 5773	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢
8	4, 7	syl6req 2089	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f1stres 5786	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10		ffn 5046	. . . . . . . . . . . . 13 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
11	9, 10	ax-mp 7	. . . . . . . . . . . 12 ⊢ (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12		fnfvelrn 5299	. . . . . . . . . . . 12 ⊢ (((1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
13	11, 12	mpan 400	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2114	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 125	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
16	15	expcom 109	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1489	. . . . . . 7 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 2951	. . . . . 6 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 5052	. . . . . . 7 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
20	9, 19	ax-mp 7	. . . . . 6 ⊢ ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
21	18, 20	jctil 295	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 2960	. . . . 5 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 137	. . . 4 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 9	jctil 295	. . 3 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 5110	. . 3 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴 ↔ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
26	24, 25	sylibr 137	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
27	2, 26	sylbir 125	1 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff set class

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 1^st c1st 5765

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-1st 5767

This theorem is referenced by: 1stconst 5842

W3C validator