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Theorem mpo2eqb 7565
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7563. (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 7436 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-mpo 7436 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
31, 2eqeq12i 2753 . . 3 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
4 eqoprab2bw 7503 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)} ↔ ∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
5 pm5.32 573 . . . . . . 7 (((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
65albii 1816 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
7 19.21v 1937 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
86, 7bitr3i 277 . . . . 5 (∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
982albii 1817 . . . 4 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
10 r2al 3193 . . . 4 (∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
119, 10bitr4i 278 . . 3 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
123, 4, 113bitri 297 . 2 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
13 pm13.183 3666 . . . . . 6 (𝐶𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1413ralimi 3081 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
15 ralbi 3101 . . . . 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 3081 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
18 ralbi 3101 . . 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 290 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  {coprab 7432  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  homfeq  17739  comfeq  17751  2arymaptf1  48503
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