Theorem abrexss 6352

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 2581	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		abrexss.1	. . . . 5 ⊢ Ⅎ𝑥𝐶
3	2	nfcri 2386	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
4		eleq1 2301	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
5		vex 2824	. . . . 5 ⊢ 𝑧 ∈ V
6	5	a1i 9	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → 𝑧 ∈ V)
7		rspa 2598	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
8	1, 3, 4, 6, 7	elabreximd 6350	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑧 ∈ 𝐶)
9	8	ex 115	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑧 ∈ 𝐶))
10	9	ssrdv 3254	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  {cab 2224  Ⅎwnfc 2379  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  funimass4f  6353