Theorem 2ndimaxp 33127

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 6077	. . . 4 ⊢ (2nd “ ∅) = ∅
2		xpeq2 5680	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3		xp0 5759	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
4	2, 3	eqtrdi 2813	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
5	4	imaeq2d 6060	. . . 4 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6		id 23	. . . 4 ⊢ (𝐵 = ∅ → 𝐵 = ∅)
7	1, 5, 6	3eqtr4a 2823	. . 3 ⊢ (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
8	7	adantl 487	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9		xpnz 6155	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10		fo2nd 8011	. . . . . . 7 ⊢ 2nd :V–onto→V
11		fofn 6795	. . . . . . 7 ⊢ (2nd :V–onto→V → 2nd Fn V)
12	10, 11	mp1i 14	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13		ssv 3958	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
15	12, 14	fvelimabd 6955	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦))
16	9, 15	sylbi 220	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦))
17		simpr 490	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) = 𝑦)
18		xp2nd 8023	. . . . . . . 8 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
19	18	ad2antlr 740	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) ∈ 𝐵)
20	17, 19	eqeltrrd 2863	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
21	20	r19.29an 3168	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵)
22		n0 4303	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
23	22	biimpi 219	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴)
24	23	ad2antrr 739	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
25		opelxpi 5696	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
26	25	ancoms 464	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
27	26	adantll 727	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28		fveqeq2 6891	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
29	28	adantl 487	. . . . . . 7 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
30		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
31		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
32	30, 31	op2nd 7999	. . . . . . . 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 3581	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
35	24, 34	exlimddv 1968	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)
36	21, 35	impbida 813	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦 ↔ 𝑦 ∈ 𝐵))
37	16, 36	bitrd 282	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ 𝑦 ∈ 𝐵))
38	37	eqrdv 2760	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
39	8, 38	pm2.61dane 3044	1 ⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  ⟨cop 4593   × cxp 5657   “ cima 5662   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  2^nd c2nd 7989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-2nd 7991

This theorem is used by:  gsumpart  33511