Theorem abrexss 32181

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 3280	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		abrexss.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfcri 2889	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
4		eleq1 2820	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		vex 3477	. . . . 5 ⊢ 𝑧 ∈ V
6	5	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → 𝑧 ∈ V)
7		rspa 3244	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
8	1, 3, 4, 6, 7	elabreximd 32179	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑧 ∈ 𝐶)
9	8	ex 412	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑧 ∈ 𝐶))
10	9	ssrdv 3988	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  {cab 2708  Ⅎwnfc 2882  ∀wral 3060  ∃wrex 3069  Vcvv 3473   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965

This theorem is referenced by:  funimass4f  32293  measvunilem  33673