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Theorem fo1stresm 6216

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 2256	. . 3 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
2	1	cbvexv 1930	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
3		opelxp 4690	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5579	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1^st ‘⟨𝑢, 𝑣⟩))
5		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op1st 6201	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢
8	4, 7	eqtr2di 2243	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f1stres 6214	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10		ffn 5404	. . . . . . . . . . . . 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 5691	. . . . . . . . . . . 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 2270	. . . . . . . . . 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 1609	. . . . . . 7 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3186	. . . . . 6 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 5413	. . . . . . 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 3195	. . . . 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 5481	. . 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 1364 ∃wex 1503 ∈ wcel 2164 ⊆ wss 3154 ⟨cop 3622 × cxp 4658 ran crn 4661 ↾ cres 4662 Fn wfn 5250 ⟶wf 5251 –onto→wfo 5253 ‘cfv 5255 1^st c1st 6193

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fo 5261 df-fv 5263 df-1st 6195

This theorem is referenced by: 1stconst 6276

W3C validator