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	⊢ Ⅎ𝑥𝐶

Assertion

Ref	Expression
abrexss	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		nfra1 3123	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		abrexss.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfcri 2929	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
4		eleq1 2847	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		vex 3401	. . . . 5 ⊢ 𝑧 ∈ V
6	5	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → 𝑧 ∈ V)
7		rspa 3112	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
8	1, 3, 4, 6, 7	elabreximd 29927	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑧 ∈ 𝐶)
9	8	ex 403	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑧 ∈ 𝐶))
10	9	ssrdv 3827	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  {cab 2763  Ⅎwnfc 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