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Theorem abrexss 32254
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 3275 . . . 4 𝑥𝑥𝐴 𝐵𝐶
2 abrexss.1 . . . . 5 𝑥𝐶
32nfcri 2884 . . . 4 𝑥 𝑧𝐶
4 eleq1 2815 . . . 4 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
5 vex 3472 . . . . 5 𝑧 ∈ V
65a1i 11 . . . 4 (∀𝑥𝐴 𝐵𝐶𝑧 ∈ V)
7 rspa 3239 . . . 4 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
81, 3, 4, 6, 7elabreximd 32252 . . 3 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑧𝐶)
98ex 412 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑧𝐶))
109ssrdv 3983 1 (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {cab 2703  wnfc 2877  wral 3055  wrex 3064  Vcvv 3468  wss 3943
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-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960
This theorem is referenced by:  funimass4f  32366  measvunilem  33740
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