Theorem 2ndimaxp 32656

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 6095	. . . 4 ⊢ (2nd “ ∅) = ∅
2		xpeq2 5706	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3		xp0 6178	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2793	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
5	4	imaeq2d 6078	. . . 4 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6		id 22	. . . 4 ⊢ (𝐵 = ∅ → 𝐵 = ∅)
7	1, 5, 6	3eqtr4a 2803	. . 3 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
8	7	adantl 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9		xpnz 6179	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10		fo2nd 8035	. . . . . . 7 ⊢ 2nd :V–onto→V
11		fofn 6822	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13		ssv 4008	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
15	12, 14	fvelimabd 6982	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦))
16	9, 15	sylbi 217	. . . 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 8047	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
19	18	ad2antlr 727	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) ∈ 𝐵)
20	17, 19	eqeltrrd 2842	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	r19.29an 3158	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
22		n0 4353	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
23	22	biimpi 216	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
24	23	ad2antrr 726	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
25		opelxpi 5722	. . . . . . . . 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 714	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28		fveqeq2 6915	. . . . . . . 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 3484	. . . . . . . . 9 ⊢ 𝑥 ∈ V
31		vex 3484	. . . . . . . . 9 ⊢ 𝑦 ∈ V
32	30, 31	op2nd 8023	. . . . . . . 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 3624	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
35	24, 34	exlimddv 1935	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
36	21, 35	impbida 801	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦 ↔ 𝑦 ∈ 𝐵))
37	16, 36	bitrd 279	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ 𝑦 ∈ 𝐵))
38	37	eqrdv 2735	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
39	8, 38	pm2.61dane 3029	1 ⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ⟨cop 4632   × cxp 5683   “ cima 5688   Fn wfn 6556  –onto→wfo 6559  ‘cfv 6561  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-2nd 8015

This theorem is referenced by:  gsumpart  33060