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Theorem bj-mpomptALT 35619
Description: Alternate proof of mpompt 7475. (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 5662 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩)
21anbi1i 625 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
3 r19.41v 3186 . . . 4 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑥𝐴𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
4 r19.41v 3186 . . . . . 6 (∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶))
5 bj-mpomptALT.1 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
65eqeq2d 2748 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑡 = 𝐶𝑡 = 𝐷))
76pm5.32i 576 . . . . . . 7 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
87rexbii 3098 . . . . . 6 (∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
94, 8bitr3i 277 . . . . 5 ((∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
109rexbii 3098 . . . 4 (∃𝑥𝐴 (∃𝑦𝐵 𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
112, 3, 103bitr2i 299 . . 3 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷))
1211opabbii 5177 . 2 {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)} = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
13 df-mpt 5194 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑡⟩ ∣ (𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑡 = 𝐶)}
14 bj-dfmpoa 35618 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑡⟩ ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝑡 = 𝐷)}
1512, 13, 143eqtr4i 2775 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3074  cop 4597  {copab 5172  cmpt 5193   × cxp 5636  cmpo 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-mpt 5194  df-xp 5644  df-oprab 7366  df-mpo 7367
This theorem is referenced by: (None)
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