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Theorem intimass 39877
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 3321 . . 3 (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
2 elimaint 39871 . . 3 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
3 elintima 39876 . . 3 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
41, 2, 33imtr4i 293 . 2 (𝑦 ∈ ( 𝐴𝐵) → 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
54ssriv 3968 1 ( 𝐴𝐵) ⊆ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  wss 3933  cop 4563   cint 4867  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  intimass2  39878
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