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Theorem abrexss 29929
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 3123 . . . 4 𝑥𝑥𝐴 𝐵𝐶
2 abrexss.1 . . . . 5 𝑥𝐶
32nfcri 2929 . . . 4 𝑥 𝑧𝐶
4 eleq1 2847 . . . 4 (𝑧 = 𝐵 → (𝑧𝐶𝐵𝐶))
5 vex 3401 . . . . 5 𝑧 ∈ V
65a1i 11 . . . 4 (∀𝑥𝐴 𝐵𝐶𝑧 ∈ V)
7 rspa 3112 . . . 4 ((∀𝑥𝐴 𝐵𝐶𝑥𝐴) → 𝐵𝐶)
81, 3, 4, 6, 7elabreximd 29927 . . 3 ((∀𝑥𝐴 𝐵𝐶𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}) → 𝑧𝐶)
98ex 403 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} → 𝑧𝐶))
109ssrdv 3827 1 (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  {cab 2763  wnfc 2919  wral 3090  wrex 3091  Vcvv 3398  wss 3792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-in 3799  df-ss 3806
This theorem is referenced by:  funimass4f  30019  measvunilem  30881
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