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Theorem fo1stres 7963

Description: Onto mapping of a restriction of the 1^st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

**Assertion**
Ref	Expression
fo1stres	⊢ (𝐵 ≠ ∅ → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Proof of Theorem fo1stres Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4294	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		opelxp 5662	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		fvres 6855	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1^st ‘⟨𝑥, 𝑦⟩))
4		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
5		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	4, 5	op1st 7945	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
7	3, 6	eqtr2di 2789	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8		f1stres 7961	. . . . . . . . . . . . 13 ⊢ (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9		ffn 6664	. . . . . . . . . . . . 13 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
11		fnfvelrn 7028	. . . . . . . . . . . 12 ⊢ (((1^st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
12	10, 11	mpan 691	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1^st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1^st ↾ (𝐴 × 𝐵)))
13	7, 12	eqeltrd 2837	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
14	2, 13	sylbir 235	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵)))
15	14	expcom 413	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
16	15	exlimiv 1932	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
17	1, 16	sylbi 217	. . . . . 6 ⊢ (𝐵 ≠ ∅ → (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (1^st ↾ (𝐴 × 𝐵))))
18	17	ssrdv 3928	. . . . 5 ⊢ (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵)))
19		frn 6671	. . . . . 6 ⊢ ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
20	8, 19	ax-mp 5	. . . . 5 ⊢ ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
21	18, 20	jctil 519	. . . 4 ⊢ (𝐵 ≠ ∅ → (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
22		eqss 3938	. . . 4 ⊢ (ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1^st ↾ (𝐴 × 𝐵)) ⊆ 𝐴 ∧ 𝐴 ⊆ ran (1^st ↾ (𝐴 × 𝐵))))
23	21, 22	sylibr 234	. . 3 ⊢ (𝐵 ≠ ∅ → ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴)
24	23, 8	jctil 519	. 2 ⊢ (𝐵 ≠ ∅ → ((1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1^st ↾ (𝐴 × 𝐵)) = 𝐴))
25		dffo2 6752	. 2 ⊢ ((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 234	1 ⊢ (𝐵 ≠ ∅ → (1^st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ⊆ wss 3890 ∅c0 4274 ⟨cop 4574 × cxp 5624 ran crn 5627 ↾ cres 5628 Fn wfn 6489 ⟶wf 6490 –onto→wfo 6492 ‘cfv 6494 1^st c1st 7935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pr 5372 ax-un 7684

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-fo 6500 df-fv 6502 df-1st 7937

This theorem is referenced by: 1stconst 8045 txcmpb 23623

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