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Theorem 2ndimaxp 32663
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 6097 . . . 4 (2nd “ ∅) = ∅
2 xpeq2 5710 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3 xp0 6180 . . . . . 6 (𝐴 × ∅) = ∅
42, 3eqtrdi 2791 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
54imaeq2d 6080 . . . 4 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6 id 22 . . . 4 (𝐵 = ∅ → 𝐵 = ∅)
71, 5, 63eqtr4a 2801 . . 3 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
87adantl 481 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9 xpnz 6181 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10 fo2nd 8034 . . . . . . 7 2nd :V–onto→V
11 fofn 6823 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
1210, 11mp1i 13 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13 ssv 4020 . . . . . . 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 8046 . . . . . . . 8 (𝑝 ∈ (𝐴 × 𝐵) → (2nd𝑝) ∈ 𝐵)
1918ad2antlr 727 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → (2nd𝑝) ∈ 𝐵)
2017, 19eqeltrrd 2840 . . . . . 6 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → 𝑦𝐵)
2120r19.29an 3156 . . . . 5 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦) → 𝑦𝐵)
22 n0 4359 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2322biimpi 216 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
2423ad2antrr 726 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐴)
25 opelxpi 5726 . . . . . . . . 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 6916 . . . . . . . 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 3482 . . . . . . . . 9 𝑥 ∈ V
31 vex 3482 . . . . . . . . 9 𝑦 ∈ V
3230, 31op2nd 8022 . . . . . . . 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 1933 . . . . 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 2733 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
398, 38pm2.61dane 3027 1 (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  wss 3963  c0 4339  cop 4637   × cxp 5687  cima 5692   Fn wfn 6558  ontowfo 6561  cfv 6563  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-2nd 8014
This theorem is referenced by:  gsumpart  33043
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