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Theorem mpo2eqb 5930
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 5928. (Contributed by Mario Carneiro, 4-Jan-2017.)
Assertion
Ref Expression
mpo2eqb (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem mpo2eqb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mpo 5829 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-mpo 5829 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
31, 2eqeq12i 2171 . . 3 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
4 eqoprab2b 5879 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)} ↔ ∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
5 pm5.32 449 . . . . . . 7 (((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
65albii 1450 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
7 19.21v 1853 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
86, 7bitr3i 185 . . . . 5 (∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
982albii 1451 . . . 4 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
10 r2al 2476 . . . 4 (∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
119, 10bitr4i 186 . . 3 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
123, 4, 113bitri 205 . 2 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
13 pm13.183 2850 . . . . . 6 (𝐶𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1413ralimi 2520 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
15 ralbi 2589 . . . . 5 (∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1614, 15syl 14 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1716ralimi 2520 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
18 ralbi 2589 . . 3 (∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1917, 18syl 14 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
2012, 19bitr4id 198 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333   = wceq 1335  wcel 2128  wral 2435  {coprab 5825  cmpo 5826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-oprab 5828  df-mpo 5829
This theorem is referenced by: (None)
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