Theorem 2ndimaxp 30885

Description: Image of a cartesian product by 2^nd. (Contributed by Thierry Arnoux, 23-Jun-2024.)

Assertion

Ref	Expression
2ndimaxp	⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)

Proof of Theorem 2ndimaxp

Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ima0 5974	. . . 4 ⊢ (2nd “ ∅) = ∅
2		xpeq2 5601	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3		xp0 6050	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2795	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
5	4	imaeq2d 5958	. . . 4 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6		id 22	. . . 4 ⊢ (𝐵 = ∅ → 𝐵 = ∅)
7	1, 5, 6	3eqtr4a 2805	. . 3 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
8	7	adantl 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9		xpnz 6051	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10		fo2nd 7825	. . . . . . 7 ⊢ 2nd :V–onto→V
11		fofn 6674	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13		ssv 3941	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
15	12, 14	fvelimabd 6824	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦))
16	9, 15	sylbi 216	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦))
17		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) = 𝑦)
18		xp2nd 7837	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
19	18	ad2antlr 723	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) ∈ 𝐵)
20	17, 19	eqeltrrd 2840	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	r19.29an 3216	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
22		n0 4277	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
23	22	biimpi 215	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
24	23	ad2antrr 722	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
25		opelxpi 5617	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
26	25	ancoms 458	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
27	26	adantll 710	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28		fveqeq2 6765	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
29	28	adantl 481	. . . . . . 7 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
30		vex 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
31		vex 3426	. . . . . . . . 9 ⊢ 𝑦 ∈ V
32	30, 31	op2nd 7813	. . . . . . . 8 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
33	32	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦)
34	27, 29, 33	rspcedvd 3555	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
35	24, 34	exlimddv 1939	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
36	21, 35	impbida 797	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦 ↔ 𝑦 ∈ 𝐵))
37	16, 36	bitrd 278	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ 𝑦 ∈ 𝐵))
38	37	eqrdv 2736	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
39	8, 38	pm2.61dane 3031	1 ⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ⟨cop 4564   × cxp 5578   “ cima 5583   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-2nd 7805

This theorem is referenced by:  gsumpart  31217