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Theorem mpteq12f 3837
 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 1434 . . . 4 𝑥𝑥 𝐴 = 𝐶
2 nfra1 2355 . . . 4 𝑥𝑥𝐴 𝐵 = 𝐷
31, 2nfan 1457 . . 3 𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
4 nfv 1421 . . 3 𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
5 rsp 2369 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐷 → (𝑥𝐴𝐵 = 𝐷))
65imp 115 . . . . . 6 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → 𝐵 = 𝐷)
76eqeq2d 2051 . . . . 5 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → (𝑦 = 𝐵𝑦 = 𝐷))
87pm5.32da 425 . . . 4 (∀𝑥𝐴 𝐵 = 𝐷 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐷)))
9 sp 1401 . . . . . 6 (∀𝑥 𝐴 = 𝐶𝐴 = 𝐶)
109eleq2d 2107 . . . . 5 (∀𝑥 𝐴 = 𝐶 → (𝑥𝐴𝑥𝐶))
1110anbi1d 438 . . . 4 (∀𝑥 𝐴 = 𝐶 → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐶𝑦 = 𝐷)))
128, 11sylan9bbr 436 . . 3 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
133, 4, 12opabbid 3822 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
14 df-mpt 3820 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
15 df-mpt 3820 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
1613, 14, 153eqtr4g 2097 1 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {copab 3817   ↦ cmpt 3818 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820 This theorem is referenced by:  mpteq12dva  3838  mpteq12  3840  mpteq2ia  3843  mpteq2da  3846
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