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Theorem mpteq12f 5146
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12f ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfa1 2148 . . . 4 𝑥𝑥 𝐴 = 𝐶
2 nfra1 3224 . . . 4 𝑥𝑥𝐴 𝐵 = 𝐷
31, 2nfan 1893 . . 3 𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
4 nfv 1908 . . 3 𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
5 rspa 3211 . . . . . 6 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → 𝐵 = 𝐷)
65eqeq2d 2837 . . . . 5 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → (𝑦 = 𝐵𝑦 = 𝐷))
76pm5.32da 579 . . . 4 (∀𝑥𝐴 𝐵 = 𝐷 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐷)))
8 sp 2174 . . . . . 6 (∀𝑥 𝐴 = 𝐶𝐴 = 𝐶)
98eleq2d 2903 . . . . 5 (∀𝑥 𝐴 = 𝐶 → (𝑥𝐴𝑥𝐶))
109anbi1d 629 . . . 4 (∀𝑥 𝐴 = 𝐶 → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐶𝑦 = 𝐷)))
117, 10sylan9bbr 511 . . 3 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
123, 4, 11opabbid 5128 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
13 df-mpt 5144 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
14 df-mpt 5144 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
1512, 13, 143eqtr4g 2886 1 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528   = wceq 1530  wcel 2107  wral 3143  {copab 5125  cmpt 5143
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-ral 3148  df-opab 5126  df-mpt 5144
This theorem is referenced by:  mpteq12dva  5147  mpteq12  5150  mpteq2ia  5154  mpteq2da  5157  esumeq12dvaf  31176  refsum2cnlem1  41159  mpteq1df  41371  mpteq12da  41378  smfsupmpt  42955  smfinflem  42957  smfinfmpt  42959
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