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Theorem 2ndimaxp 32656
Description: Image of a cartesian product by 2nd. (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.
StepHypRef Expression
1 ima0 6095 . . . 4 (2nd “ ∅) = ∅
2 xpeq2 5706 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3 xp0 6178 . . . . . 6 (𝐴 × ∅) = ∅
42, 3eqtrdi 2793 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
54imaeq2d 6078 . . . 4 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6 id 22 . . . 4 (𝐵 = ∅ → 𝐵 = ∅)
71, 5, 63eqtr4a 2803 . . 3 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
87adantl 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)
1210, 11mp1i 13 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13 ssv 4008 . . . . . . 7 (𝐴 × 𝐵) ⊆ V
1413a1i 11 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
1512, 14fvelimabd 6982 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦))
169, 15sylbi 217 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦))
17 simpr 484 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → (2nd𝑝) = 𝑦)
18 xp2nd 8047 . . . . . . . 8 (𝑝 ∈ (𝐴 × 𝐵) → (2nd𝑝) ∈ 𝐵)
1918ad2antlr 727 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → (2nd𝑝) ∈ 𝐵)
2017, 19eqeltrrd 2842 . . . . . 6 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → 𝑦𝐵)
2120r19.29an 3158 . . . . 5 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦) → 𝑦𝐵)
22 n0 4353 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2322biimpi 216 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
2423ad2antrr 726 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐴)
25 opelxpi 5722 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2625ancoms 458 . . . . . . . 8 ((𝑦𝐵𝑥𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2726adantll 714 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28 fveqeq2 6915 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → ((2nd𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
2928adantl 481 . . . . . . 7 (((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((2nd𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
30 vex 3484 . . . . . . . . 9 𝑥 ∈ V
31 vex 3484 . . . . . . . . 9 𝑦 ∈ V
3230, 31op2nd 8023 . . . . . . . 8 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
3332a1i 11 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦)
3427, 29, 33rspcedvd 3624 . . . . . 6 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦)
3524, 34exlimddv 1935 . . . . 5 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦)
3621, 35impbida 801 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦𝑦𝐵))
3716, 36bitrd 279 . . 3 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ 𝑦𝐵))
3837eqrdv 2735 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
398, 38pm2.61dane 3029 1 (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
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  ontowfo 6559  cfv 6561  2nd 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
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