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Theorem mpo2eqb 7537
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7535. (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 7410 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 df-mpo 7410 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
31, 2eqeq12i 2744 . . 3 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
4 eqoprab2bw 7475 . . 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 1813 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
7 19.21v 1934 . . . . . 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 1814 . . . 4 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
10 r2al 3188 . . . 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 3651 . . . . . 6 (𝐶𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1413ralimi 3077 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
15 ralbi 3097 . . . . 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 3077 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
18 ralbi 3097 . . 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 205  wa 395  wal 1531   = wceq 1533  wcel 2098  wral 3055  {coprab 7406  cmpo 7407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  homfeq  17647  comfeq  17659  2arymaptf1  47611
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