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Theorem fo1stresm 6152

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 2238	. . 3 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
2	1	cbvexv 1916	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
3		opelxp 4650	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5531	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1^st ‘⟨𝑢, 𝑣⟩))
5		vex 2738	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2738	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op1st 6137	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢
8	4, 7	eqtr2di 2225	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f1stres 6150	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10		ffn 5357	. . . . . . . . . . . . 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 5640	. . . . . . . . . . . 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 424	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2252	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 135	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
16	15	expcom 116	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1596	. . . . . . 7 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3159	. . . . . 6 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 5366	. . . . . . 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 312	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3168	. . . . 5 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 134	. . . 4 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 9	jctil 312	. . 3 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 5434	. . 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 134	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
27	2, 26	sylbir 135	1 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1490 ∈ wcel 2146 ⊆ wss 3127 ⟨cop 3592 × cxp 4618 ran crn 4621 ↾ cres 4622 Fn wfn 5203 ⟶wf 5204 –onto→wfo 5206 ‘cfv 5208 1^st c1st 6129

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 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fo 5214 df-fv 5216 df-1st 6131

This theorem is referenced by: 1stconst 6212

W3C validator