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Theorem 2ndimaxp 30412
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 5916 . . . 4 (2nd “ ∅) = ∅
2 xpeq2 5544 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
3 xp0 5986 . . . . . 6 (𝐴 × ∅) = ∅
42, 3eqtrdi 2852 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
54imaeq2d 5900 . . . 4 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = (2nd “ ∅))
6 id 22 . . . 4 (𝐵 = ∅ → 𝐵 = ∅)
71, 5, 63eqtr4a 2862 . . 3 (𝐵 = ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
87adantl 485 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
9 xpnz 5987 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
10 fo2nd 7696 . . . . . . 7 2nd :V–onto→V
11 fofn 6571 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
1210, 11mp1i 13 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn V)
13 ssv 3942 . . . . . . 7 (𝐴 × 𝐵) ⊆ V
1413a1i 11 . . . . . 6 ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V)
1512, 14fvelimabd 6717 . . . . 5 ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦))
169, 15sylbi 220 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦))
17 simpr 488 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → (2nd𝑝) = 𝑦)
18 xp2nd 7708 . . . . . . . 8 (𝑝 ∈ (𝐴 × 𝐵) → (2nd𝑝) ∈ 𝐵)
1918ad2antlr 726 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → (2nd𝑝) ∈ 𝐵)
2017, 19eqeltrrd 2894 . . . . . 6 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd𝑝) = 𝑦) → 𝑦𝐵)
2120r19.29an 3250 . . . . 5 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦) → 𝑦𝐵)
22 n0 4263 . . . . . . . 8 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2322biimpi 219 . . . . . . 7 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
2423ad2antrr 725 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) → ∃𝑥 𝑥𝐴)
25 opelxpi 5560 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2625ancoms 462 . . . . . . . 8 ((𝑦𝐵𝑥𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2726adantll 713 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
28 fveqeq2 6658 . . . . . . . 8 (𝑝 = ⟨𝑥, 𝑦⟩ → ((2nd𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
2928adantl 485 . . . . . . 7 (((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) ∧ 𝑝 = ⟨𝑥, 𝑦⟩) → ((2nd𝑝) = 𝑦 ↔ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦))
30 vex 3447 . . . . . . . . 9 𝑥 ∈ V
31 vex 3447 . . . . . . . . 9 𝑦 ∈ V
3230, 31op2nd 7684 . . . . . . . 8 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
3332a1i 11 . . . . . . 7 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦)
3427, 29, 33rspcedvd 3577 . . . . . 6 ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) ∧ 𝑥𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦)
3524, 34exlimddv 1936 . . . . 5 (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦)
3621, 35impbida 800 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (∃𝑝 ∈ (𝐴 × 𝐵)(2nd𝑝) = 𝑦𝑦𝐵))
3716, 36bitrd 282 . . 3 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ 𝑦𝐵))
3837eqrdv 2799 . 2 ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (2nd “ (𝐴 × 𝐵)) = 𝐵)
398, 38pm2.61dane 3077 1 (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wne 2990  wrex 3110  Vcvv 3444  wss 3884  c0 4246  cop 4534   × cxp 5521  cima 5526   Fn wfn 6323  ontowfo 6326  cfv 6328  2nd c2nd 7674
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-2nd 7676
This theorem is referenced by:  gsumpart  30743
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