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Theorem mpoeq123dva 7427
Description: An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpoeq123dv.1 (𝜑𝐴 = 𝐷)
mpoeq123dva.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
mpoeq123dva.3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
Assertion
Ref Expression
mpoeq123dva (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 mpoeq123dva
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mpoeq123dva.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐹)
21eqeq2d 2747 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = 𝐶𝑧 = 𝐹))
32pm5.32da 579 . . . 4 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐹)))
4 mpoeq123dva.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 = 𝐸)
54eleq2d 2823 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑦𝐵𝑦𝐸))
65pm5.32da 579 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴𝑦𝐸)))
7 mpoeq123dv.1 . . . . . . . 8 (𝜑𝐴 = 𝐷)
87eleq2d 2823 . . . . . . 7 (𝜑 → (𝑥𝐴𝑥𝐷))
98anbi1d 630 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐸) ↔ (𝑥𝐷𝑦𝐸)))
106, 9bitrd 278 . . . . 5 (𝜑 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐷𝑦𝐸)))
1110anbi1d 630 . . . 4 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐹) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
123, 11bitrd 278 . . 3 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)))
1312oprabbidv 7419 . 2 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)})
14 df-mpo 7358 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
15 df-mpo 7358 . 2 (𝑥𝐷, 𝑦𝐸𝐹) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐷𝑦𝐸) ∧ 𝑧 = 𝐹)}
1613, 14, 153eqtr4g 2801 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐷, 𝑦𝐸𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {coprab 7354  cmpo 7355
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-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-oprab 7357  df-mpo 7358
This theorem is referenced by:  mpoeq123dv  7428  natpropd  17857  fucpropd  17858  curfpropd  18114  hofpropd  18148  rrxdsfi  24759  istrkgl  27286  eengv  27814  elntg  27819  submat1n  32255  rrxtopnfi  44460  rngcifuestrc  46227  funcrngcsetc  46228  funcrngcsetcALT  46229  funcringcsetc  46265  eenglngeehlnm  46757
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