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Theorem mptssALT 30548
 Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5887. (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 3887 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 613 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑦 = 𝐶) → (𝑥𝐵𝑦 = 𝐶)))
32ssopab2dv 5412 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
4 df-mpt 5117 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
5 df-mpt 5117 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
63, 4, 53sstr4g 3939 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  {copab 5098   ↦ cmpt 5116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-opab 5099  df-mpt 5117 This theorem is referenced by: (None)
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