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Theorem mpteq12f 4098
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 1552 . . . 4 𝑥𝑥 𝐴 = 𝐶
2 nfra1 2521 . . . 4 𝑥𝑥𝐴 𝐵 = 𝐷
31, 2nfan 1576 . . 3 𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
4 nfv 1539 . . 3 𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
5 rsp 2537 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐷 → (𝑥𝐴𝐵 = 𝐷))
65imp 124 . . . . . 6 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → 𝐵 = 𝐷)
76eqeq2d 2201 . . . . 5 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → (𝑦 = 𝐵𝑦 = 𝐷))
87pm5.32da 452 . . . 4 (∀𝑥𝐴 𝐵 = 𝐷 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐷)))
9 sp 1522 . . . . . 6 (∀𝑥 𝐴 = 𝐶𝐴 = 𝐶)
109eleq2d 2259 . . . . 5 (∀𝑥 𝐴 = 𝐶 → (𝑥𝐴𝑥𝐶))
1110anbi1d 465 . . . 4 (∀𝑥 𝐴 = 𝐶 → ((𝑥𝐴𝑦 = 𝐷) ↔ (𝑥𝐶𝑦 = 𝐷)))
128, 11sylan9bbr 463 . . 3 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
133, 4, 12opabbid 4083 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
14 df-mpt 4081 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
15 df-mpt 4081 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
1613, 14, 153eqtr4g 2247 1 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362   = wceq 1364  wcel 2160  wral 2468  {copab 4078  cmpt 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-opab 4080  df-mpt 4081
This theorem is referenced by:  mpteq12dva  4099  mpteq12  4101  mpteq2ia  4104  mpteq2da  4107
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