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Theorem abrexss 30282
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 3213 . . . 4 𝑥𝑥𝐴 𝐵𝐶
2 abrexss.1 . . . . 5 𝑥𝐶
32nfcri 2969 . . . 4 𝑥 𝑧𝐶
4 eleq1 2903 . . . 4 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
5 vex 3483 . . . . 5 𝑧 ∈ V
65a1i 11 . . . 4 (∀𝑥𝐴 𝐵𝐶𝑧 ∈ V)
7 rspa 3201 . . . 4 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
81, 3, 4, 6, 7elabreximd 30280 . . 3 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑧𝐶)
98ex 416 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑧𝐶))
109ssrdv 3959 1 (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {cab 2802  wnfc 2962  wral 3133  wrex 3134  Vcvv 3480  wss 3919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936
This theorem is referenced by:  funimass4f  30393  measvunilem  31528
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