Theorem 2ndimaxp 32735

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 6044	. . . 4 ⊢ (2nd “ ∅) = ∅
2		xpeq2 5653	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3		xp0 5732	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2788	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
5	4	imaeq2d 6027	. . . 4 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6		id 22	. . . 4 ⊢ (𝐵 = ∅ → 𝐵 = ∅)
7	1, 5, 6	3eqtr4a 2798	. . 3 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
8	7	adantl 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9		xpnz 6125	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10		fo2nd 7964	. . . . . . 7 ⊢ 2nd :V–onto→V
11		fofn 6756	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
12	10, 11	mp1i 13	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13		ssv 3960	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
15	12, 14	fvelimabd 6915	. . . . 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 7976	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
19	18	ad2antlr 728	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) ∈ 𝐵)
20	17, 19	eqeltrrd 2838	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	r19.29an 3142	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
22		n0 4307	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
23	22	biimpi 216	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
24	23	ad2antrr 727	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
25		opelxpi 5669	. . . . . . . . 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 715	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28		fveqeq2 6851	. . . . . . . 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 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
31		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
32	30, 31	op2nd 7952	. . . . . . . 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 3580	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
35	24, 34	exlimddv 1937	. . . . 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 3020	1 ⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ⟨cop 4588   × cxp 5630   “ cima 5635   Fn wfn 6495  –onto→wfo 6498  ‘cfv 6500  2^nd c2nd 7942

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 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-2nd 7944

This theorem is referenced by:  gsumpart  33156