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Theorem bj-mpomptALT 36092
Description: Alternate proof of mpompt 7524. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
bj-mpomptALT.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
bj-mpomptALT (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷
Allowed substitution hints:   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem bj-mpomptALT
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 elxp2 5700 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
21anbi1i 624 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
3 r19.41v 3188 . . . 4 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
4 r19.41v 3188 . . . . . 6 (∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
5 bj-mpomptALT.1 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
65eqeq2d 2743 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑡 = 𝐶𝑡 = 𝐷))
76pm5.32i 575 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
87rexbii 3094 . . . . . 6 (∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
94, 8bitr3i 276 . . . . 5 ((∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
109rexbii 3094 . . . 4 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
112, 3, 103bitr2i 298 . . 3 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
1211opabbii 5215 . 2 {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
13 df-mpt 5232 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)}
14 bj-dfmpoa 36091 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
1512, 13, 143eqtr4i 2770 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  cop 4634  {copab 5210  cmpt 5231   × cxp 5674  cmpo 7413
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-mpt 5232  df-xp 5682  df-oprab 7415  df-mpo 7416
This theorem is referenced by: (None)
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