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Theorem intimag 38719
Description: Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
intimag (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
Distinct variable groups:   𝑥,𝑎,𝑦,𝐴   𝐵,𝑎,𝑥,𝑦   𝐴,𝑏   𝐵,𝑏,𝑎,𝑦,𝑥

Proof of Theorem intimag
StepHypRef Expression
1 r19.12 3242 . . . . 5 (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 → ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
2 id 22 . . . . 5 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎))
31, 2impbid2 218 . . . 4 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎))
4 elimaint 38711 . . . 4 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
5 elintima 38716 . . . 4 (𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)
63, 4, 53bitr4g 306 . . 3 ((∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → (𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
76alimi 1907 . 2 (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ∀𝑦(𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
8 dfcleq 2791 . 2 (( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑦(𝑦 ∈ ( 𝐴𝐵) ↔ 𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}))
97, 8sylibr 226 1 (∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651   = wceq 1653  wcel 2157  {cab 2783  wral 3087  wrex 3088  cop 4372   cint 4665  cima 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-br 4842  df-opab 4904  df-xp 5316  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323
This theorem is referenced by:  intimasn  38720
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