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Theorem mptssALT 28857
 Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5373. (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 3562 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 586 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑦 = 𝐶) → (𝑥𝐵𝑦 = 𝐶)))
32ssopab2dv 4929 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
4 df-mpt 4645 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
5 df-mpt 4645 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
63, 4, 53sstr4g 3609 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {copab 4642   ↦ cmpt 4643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-opab 4644  df-mpt 4645 This theorem is referenced by: (None)
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