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Theorem mpteq12f 4080
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 1541 . . . 4 𝑥𝑥 𝐴 = 𝐶
2 nfra1 2508 . . . 4 𝑥𝑥𝐴 𝐵 = 𝐷
31, 2nfan 1565 . . 3 𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
4 nfv 1528 . . 3 𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷)
5 rsp 2524 . . . . . . 7 (∀𝑥𝐴 𝐵 = 𝐷 → (𝑥𝐴𝐵 = 𝐷))
65imp 124 . . . . . 6 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → 𝐵 = 𝐷)
76eqeq2d 2189 . . . . 5 ((∀𝑥𝐴 𝐵 = 𝐷𝑥𝐴) → (𝑦 = 𝐵𝑦 = 𝐷))
87pm5.32da 452 . . . 4 (∀𝑥𝐴 𝐵 = 𝐷 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐷)))
9 sp 1511 . . . . . 6 (∀𝑥 𝐴 = 𝐶𝐴 = 𝐶)
109eleq2d 2247 . . . . 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 4065 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
14 df-mpt 4063 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
15 df-mpt 4063 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
1613, 14, 153eqtr4g 2235 1 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wcel 2148  wral 2455  {copab 4060  cmpt 4061
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-opab 4062  df-mpt 4063
This theorem is referenced by:  mpteq12dva  4081  mpteq12  4083  mpteq2ia  4086  mpteq2da  4089
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