fo1stresm - Intuitionistic Logic Explorer

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Theorem fo1stresm 5808

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 2141	. . 3 ⊢ (𝑣 = 𝑦 → (𝑣 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
2	1	cbvexv 1836	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
3		opelxp 4392	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))
4		fvres 5219	. . . . . . . . . . . 12 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1^st ‘⟨𝑢, 𝑣⟩))
5		vex 2604	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
6		vex 2604	. . . . . . . . . . . . 13 ⊢ 𝑣 ∈ V
7	5, 6	op1st 5793	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢
8	4, 7	syl6req 2130	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9		f1stres 5806	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10		ffn 5066	. . . . . . . . . . . . 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 5320	. . . . . . . . . . . 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 414	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	8, 13	eqeltrd 2155	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	3, 14	sylbir 133	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
16	15	expcom 114	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	16	exlimiv 1529	. . . . . . 7 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (𝑢 ∈ 𝐴 → 𝑢 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3005	. . . . . 6 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 5072	. . . . . . 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 305	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3014	. . . . 5 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 132	. . . 4 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 9	jctil 305	. . 3 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 5130	. . 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 132	. 2 ⊢ (∃𝑣 𝑣 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)
27	2, 26	sylbir 133	1 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ⊆ wss 2973 ⟨cop 3401 × cxp 4361 ran crn 4364 ↾ cres 4365 Fn wfn 4917 ⟶wf 4918 –onto→wfo 4920 ‘cfv 4922 1^st c1st 5785

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787

This theorem is referenced by: 1stconst 5862

W3C validator