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Theorem intimass 38263
Description: The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimass ( 𝐴𝐵) ⊆ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
Distinct variable groups:   𝑥,𝑎,𝐴   𝐵,𝑎,𝑥

Proof of Theorem intimass
Dummy variables 𝑦 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.12 3092 . . 3 (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
2 elimaint 38257 . . 3 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
3 elintima 38262 . . 3 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
41, 2, 33imtr4i 281 . 2 (𝑦 ∈ ( 𝐴𝐵) → 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
54ssriv 3640 1 ( 𝐴𝐵) ⊆ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  wss 3607  cop 4216   cint 4507  cima 5146
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  intimass2  38264
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