Theorem vsmrep 36009

Description: The universe is a standard model of the strong Axiom Scheme of Replacement. (Contributed by BTernaryTau, 2-Oct-2026.)

Assertion

Ref	Expression
vsmrep	⊢ V ⊨ Ax-Rep*

Proof of Theorem vsmrep

Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-smrep 36003	. 2 ⊢ (V ⊨ Ax-Rep* ↔ ∀𝑓∀𝑥 ∈ V (∀𝑤 ∈ V ∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ V ∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))
2		ax-rep 5231	. . . 4 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
3		ralv 3476	. . . . 5 ⊢ (∀𝑤 ∈ V ∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
4		rexv 3477	. . . . . . 7 ⊢ (∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
5		ralv 3476	. . . . . . . 8 ⊢ (∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
6	5	exbii 1880	. . . . . . 7 ⊢ (∃𝑦∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
7	4, 6	bitri 278	. . . . . 6 ⊢ (∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∃𝑦∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
8	7	albii 1851	. . . . 5 ⊢ (∀𝑤∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
9	3, 8	bitri 278	. . . 4 ⊢ (∀𝑤 ∈ V ∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) ↔ ∀𝑤∃𝑦∀𝑧(∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦))
10		rexv 3477	. . . . 5 ⊢ (∃𝑦 ∈ V ∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∃𝑦∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
11		ralv 3476	. . . . . . 7 ⊢ (∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
12		rexv 3477	. . . . . . . . 9 ⊢ (∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))
13	12	bibi2i 340	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
14	13	albii 1851	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
15	11, 14	bitri 278	. . . . . 6 ⊢ (∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
16	15	exbii 1880	. . . . 5 ⊢ (∃𝑦∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
17	10, 16	bitri 278	. . . 4 ⊢ (∃𝑦 ∈ V ∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
18	2, 9, 17	3imtr4i 295	. . 3 ⊢ (∀𝑤 ∈ V ∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ V ∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
19	18	rgenw 3080	. 2 ⊢ ∀𝑥 ∈ V (∀𝑤 ∈ V ∃𝑦 ∈ V ∀𝑧 ∈ V (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ V ∀𝑧 ∈ V (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ V (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧))))
20	1, 19	mpgbir 1831	1 ⊢ V ⊨ Ax-Rep*

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450  dom cdm 5650  Fun wfun 6532  ‘cfv 6538   ⊨ wsmrep 36001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-rep 5231

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-smrep 36003

This theorem is used by: (None)