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Theorem mpo2eqb 7278
 Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7276. (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 7155 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-mpo 7155 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
31, 2eqeq12i 2773 . . 3 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
4 eqoprab2bw 7218 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)} ↔ ∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
5 pm5.32 577 . . . . . . 7 (((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
65albii 1821 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
7 19.21v 1940 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
86, 7bitr3i 280 . . . . 5 (∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
982albii 1822 . . . 4 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
10 r2al 3130 . . . 4 (∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
119, 10bitr4i 281 . . 3 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
123, 4, 113bitri 300 . 2 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
13 pm13.183 3579 . . . . . 6 (𝐶𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1413ralimi 3092 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
15 ralbi 3099 . . . . 5 (∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1614, 15syl 17 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1716ralimi 3092 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
18 ralbi 3099 . . 3 (∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1917, 18syl 17 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
2012, 19bitr4id 293 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {coprab 7151   ∈ cmpo 7152 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-oprab 7154  df-mpo 7155 This theorem is referenced by:  homfeq  17022  comfeq  17034  2arymaptf1  45432
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