Theorem replem 5253

Description: A lemma for variants of the axiom of replacement: if we can form the set of images of the functional relation, then we can also form a set containing all its images. The converse requires the axiom of separation. (Contributed by BJ, 5-Apr-2026.)

Assertion

Ref	Expression
replem	⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Distinct variable groups:   𝜑,𝑤   𝑥,𝑤,𝑦   𝑧,𝑤,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem replem

Step	Hyp	Ref	Expression
1		biimpr 223	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → (∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤))
2		r19.23v 3198	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤) ↔ (∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤))
3	2	biimpri 231	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤) → ∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤))
4		ancr 555	. . . . . . . 8 ⊢ ((𝜑 → 𝑦 ∈ 𝑤) → (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
5	4	ralimi 3108	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤) → ∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
6	1, 3, 5	3syl 19	. . . . . 6 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
7	6	alimi 1838	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
8		ralcom4 3297	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
9	8	biimpri 231	. . . . 5 ⊢ (∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
10		exim 1861	. . . . . . 7 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → (∃𝑦𝜑 → ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑)))
11		df-rex 3096	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑))
12	10, 11	imbitrrdi 255	. . . . . 6 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
13	12	ralimi 3108	. . . . 5 ⊢ (∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
14	7, 9, 13	3syl 19	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
15		pm2.27 43	. . . . 5 ⊢ (∃𝑦𝜑 → ((∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → ∃𝑦 ∈ 𝑤 𝜑))
16	15	ral2imi 3110	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
17	14, 16	syl5 35	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
18	17	eximdv 1944	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
19	18	imp 411	1 ⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806  ∀wral 3085  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-11 2198

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-ral 3086  df-rex 3096

This theorem is referenced by:  zfrep6  5254