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Theorem abrexss 32531
Description: A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
Hypothesis
Ref Expression
abrexss.1 𝑥𝐶
Assertion
Ref Expression
abrexss (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem abrexss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfra1 3285 . . . 4 𝑥𝑥𝐴 𝐵𝐶
2 abrexss.1 . . . . 5 𝑥𝐶
32nfcri 2895 . . . 4 𝑥 𝑧𝐶
4 eleq1 2826 . . . 4 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
5 vex 3486 . . . . 5 𝑧 ∈ V
65a1i 11 . . . 4 (∀𝑥𝐴 𝐵𝐶𝑧 ∈ V)
7 rspa 3249 . . . 4 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
81, 3, 4, 6, 7elabreximd 32529 . . 3 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑧𝐶)
98ex 412 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑧𝐶))
109ssrdv 4008 1 (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  {cab 2711  wnfc 2888  wral 3063  wrex 3072  Vcvv 3482  wss 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-v 3484  df-ss 3987
This theorem is referenced by:  funimass4f  32647  measvunilem  34168
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