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Theorem mpteq12f 4890
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 2193 . . . 4 𝑥𝑥 𝐴 = 𝐶
2 nfra1 3088 . . . 4 𝑥𝑥𝐴 𝐵 = 𝐷
31, 2nfan 1998 . . 3 𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
4 nfv 2009 . . 3 𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
5 rspa 3077 . . . . . 6 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → 𝐵 = 𝐷)
65eqeq2d 2775 . . . . 5 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → (𝑦 = 𝐵𝑦 = 𝐷))
76pm5.32da 574 . . . 4 (∀𝑥𝐴 𝐵 = 𝐷 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐷)))
8 sp 2215 . . . . . 6 (∀𝑥 𝐴 = 𝐶𝐴 = 𝐶)
98eleq2d 2830 . . . . 5 (∀𝑥 𝐴 = 𝐶 → (𝑥𝐴𝑥𝐶))
109anbi1d 623 . . . 4 (∀𝑥 𝐴 = 𝐶 → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐶𝑦 = 𝐷)))
117, 10sylan9bbr 506 . . 3 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
123, 4, 11opabbid 4874 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
13 df-mpt 4889 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
14 df-mpt 4889 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
1512, 13, 143eqtr4g 2824 1 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650   = wceq 1652  wcel 2155  wral 3055  {copab 4871  cmpt 4888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-ral 3060  df-opab 4872  df-mpt 4889
This theorem is referenced by:  mpteq12dva  4891  mpteq12  4895  mpteq2ia  4899  mpteq2da  4902  esumeq12dvaf  30540  refsum2cnlem1  39848  mpteq1df  40085  mpteq12da  40094  smfsupmpt  41661  smfinflem  41663  smfinfmpt  41665
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