Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mptssALT Structured version   Visualization version   GIF version

Theorem mptssALT 32490
Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 6051. (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 3975 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 609 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑦 = 𝐶) → (𝑥𝐵𝑦 = 𝐶)))
32ssopab2dv 5557 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
4 df-mpt 5236 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
5 df-mpt 5236 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
63, 4, 53sstr4g 4027 1 (𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3949  {copab 5214  cmpt 5235
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-mpt 5236
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator