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Theorem mptssALT 29602
Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5489. (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 3630 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 587 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑦 = 𝐶) → (𝑥𝐵𝑦 = 𝐶)))
32ssopab2dv 5033 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
4 df-mpt 4763 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
5 df-mpt 4763 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
63, 4, 53sstr4g 3679 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wss 3607  {copab 4745  cmpt 4762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-in 3614  df-ss 3621  df-opab 4746  df-mpt 4763
This theorem is referenced by: (None)
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