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Theorem mptssALT 30338
 Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5909. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mptssALT (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mptssALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3965 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 610 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑦 = 𝐶) → (𝑥𝐵𝑦 = 𝐶)))
32ssopab2dv 5435 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
4 df-mpt 5144 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
5 df-mpt 5144 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
63, 4, 53sstr4g 4016 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ⊆ wss 3940  {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-11 2153  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-nfc 2968  df-in 3947  df-ss 3956  df-opab 5126  df-mpt 5144 This theorem is referenced by: (None)
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