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Theorem mpt22eqb 6734
Description: Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 6732. (Contributed by Mario Carneiro, 4-Jan-2017.)
Assertion
Ref Expression
mpt22eqb (∀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 mpt22eqb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pm13.183 3332 . . . . . 6 (𝐶𝑉 → (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
21ralimi 2948 . . . . 5 (∀𝑦𝐵 𝐶𝑉 → ∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
3 ralbi 3063 . . . . 5 (∀𝑦𝐵 (𝐶 = 𝐷 ↔ ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
42, 3syl 17 . . . 4 (∀𝑦𝐵 𝐶𝑉 → (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
54ralimi 2948 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
6 ralbi 3063 . . 3 (∀𝑥𝐴 (∀𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)) → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
75, 6syl 17 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → (∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷 ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
8 df-mpt2 6620 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 df-mpt2 6620 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
108, 9eqeq12i 2635 . . 3 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
11 eqoprab2b 6678 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)} ↔ ∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
12 pm5.32 667 . . . . . . 7 (((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
1312albii 1744 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
14 19.21v 1865 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧 = 𝐶𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1513, 14bitr3i 266 . . . . 5 (∀𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
16152albii 1745 . . . 4 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
17 r2al 2935 . . . 4 (∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧 = 𝐶𝑧 = 𝐷)))
1816, 17bitr4i 267 . . 3 (∀𝑥𝑦𝑧(((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
1910, 11, 183bitri 286 . 2 ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑧 = 𝐶𝑧 = 𝐷))
207, 19syl6rbbr 279 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑉 → ((𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷) ↔ ∀𝑥𝐴𝑦𝐵 𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2908  {coprab 6616  cmpt2 6617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  homfeq  16294  comfeq  16306
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