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Theorem mpt2eq123dva 5692
Description: An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpt2eq123dv.1 (𝜑𝐴 = 𝐷)
mpt2eq123dva.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
mpt2eq123dva.3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
Assertion
Ref Expression
mpt2eq123dva (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpt2eq123dva
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mpt2eq123dva.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
21eqeq2d 2099 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = 𝐶𝑧 = 𝐹))
32pm5.32da 440 . . . 4 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐹)))
4 mpt2eq123dva.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
54eleq2d 2157 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦𝐵𝑦𝐸))
65pm5.32da 440 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴𝑦𝐸)))
7 mpt2eq123dv.1 . . . . . . . 8 (𝜑𝐴 = 𝐷)
87eleq2d 2157 . . . . . . 7 (𝜑 → (𝑥𝐴𝑥𝐷))
98anbi1d 453 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐸) ↔ (𝑥𝐷𝑦𝐸)))
106, 9bitrd 186 . . . . 5 (𝜑 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐷𝑦𝐸)))
1110anbi1d 453 . . . 4 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐹) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
123, 11bitrd 186 . . 3 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
1312oprabbidv 5685 . 2 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
14 df-mpt2 5639 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
15 df-mpt2 5639 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
1613, 14, 153eqtr4g 2145 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  {coprab 5635  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-oprab 5638  df-mpt2 5639
This theorem is referenced by:  mpt2eq123dv  5693
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