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Theorem fo1stresm 6052

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 2200	. . 3 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
2	1	cbvexv 1890	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
3		opelxp 4564	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5438	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1^st ‘⟨𝑢, 𝑣⟩))
5		vex 2684	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2684	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op1st 6037	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢
8	4, 7	syl6req 2187	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f1stres 6050	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10		ffn 5267	. . . . . . . . . . . . 13 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
11	9, 10	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12		fnfvelrn 5545	. . . . . . . . . . . 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 420	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2214	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 134	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
16	15	expcom 115	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1577	. . . . . . 7 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3098	. . . . . 6 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 5276	. . . . . . 7 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
20	9, 19	ax-mp 5	. . . . . 6 ⊢ ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
21	18, 20	jctil 310	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3107	. . . . 5 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 133	. . . 4 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 9	jctil 310	. . 3 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 5344	. . 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 133	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
27	2, 26	sylbir 134	1 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ⊆ wss 3066 ⟨cop 3525 × cxp 4532 ran crn 4535 ↾ cres 4536 Fn wfn 5113 ⟶wf 5114 –onto→wfo 5116 ‘cfv 5118 1^st c1st 6029

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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-1st 6031

This theorem is referenced by: 1stconst 6111

W3C validator