Theorem replem 5213

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 222	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → (∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤))
2		r19.23v 3168	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤) ↔ (∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤))
3	2	biimpri 230	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝑧 𝜑 → 𝑦 ∈ 𝑤) → ∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤))
4		ancr 552	. . . . . . . 8 ⊢ ((𝜑 → 𝑦 ∈ 𝑤) → (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
5	4	ralimi 3078	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑧 (𝜑 → 𝑦 ∈ 𝑤) → ∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
6	1, 3, 5	3syl 18	. . . . . 6 ⊢ ((𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
7	6	alimi 1819	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
8		ralcom4 3267	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) ↔ ∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
9	8	biimpri 230	. . . . 5 ⊢ (∀𝑦∀𝑥 ∈ 𝑧 (𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)))
10		exim 1842	. . . . . . 7 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → (∃𝑦𝜑 → ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑)))
11		df-rex 3066	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑤 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝑤 ∧ 𝜑))
12	10, 11	imbitrrdi 254	. . . . . 6 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
13	12	ralimi 3078	. . . . 5 ⊢ (∀𝑥 ∈ 𝑧 ∀𝑦(𝜑 → (𝑦 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
14	7, 9, 13	3syl 18	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑))
15		pm2.27 42	. . . . 5 ⊢ (∃𝑦𝜑 → ((∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → ∃𝑦 ∈ 𝑤 𝜑))
16	15	ral2imi 3080	. . . 4 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
17	14, 16	syl5 34	. . 3 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
18	17	eximdv 1925	. 2 ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑))
19	18	imp 408	1 ⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546  ∃wex 1787  ∀wral 3055  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-11 2170

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-ral 3056  df-rex 3066

This theorem is referenced by:  zfrep6  5214