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Theorem abrexss 28734
 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 2925 . . . 4 𝑥𝑥𝐴 𝐵𝐶
2 abrexss.1 . . . . 5 𝑥𝐶
32nfcri 2745 . . . 4 𝑥 𝑧𝐶
4 eleq1 2676 . . . 4 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
5 vex 3176 . . . . 5 𝑧 ∈ V
65a1i 11 . . . 4 (∀𝑥𝐴 𝐵𝐶𝑧 ∈ V)
7 rspa 2914 . . . 4 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
81, 3, 4, 6, 7elabreximd 28732 . . 3 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑧𝐶)
98ex 449 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑧𝐶))
109ssrdv 3574 1 (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  funimass4f  28818  measvunilem  29602
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