Theorem wfaxrep 46027

Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5231. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in ∀𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.)

Hypothesis

Ref	Expression
wfax.1	⊢ 𝑊 = ∪ (𝑅1 “ On)

Assertion

Ref	Expression
wfaxrep	⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑊

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

Proof of Theorem wfaxrep

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfax.1	. 2 ⊢ 𝑊 = ∪ (𝑅1 “ On)
2		trwf 45992	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
3		treq 5218	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	mpbiri 261	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → Tr 𝑊)
5		vex 3454	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6	5	rnex 7922	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
7	6	r1elss 9814	. . . . . . . 8 ⊢ (ran 𝑓 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On))
8	7	biimpri 231	. . . . . . 7 ⊢ (ran 𝑓 ⊆ ∪ (𝑅1 “ On) → ran 𝑓 ∈ ∪ (𝑅1 “ On))
9	1	sseq2i 3937	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝑊 ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On))
10	1	eleq2i 2852	. . . . . . 7 ⊢ (ran 𝑓 ∈ 𝑊 ↔ ran 𝑓 ∈ ∪ (𝑅1 “ On))
11	8, 9, 10	3imtr4i 295	. . . . . 6 ⊢ (ran 𝑓 ⊆ 𝑊 → ran 𝑓 ∈ 𝑊)
12	11	3ad2ant3 1153	. . . . 5 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊)
13	12	ax-gen 1827	. . . 4 ⊢ ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊)
14	13	a1i 11	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊))
15		onwf 9840	. . . . 5 ⊢ On ⊆ ∪ (𝑅1 “ On)
16		0elon 6418	. . . . 5 ⊢ ∅ ∈ On
17	15, 16	sselii 3905	. . . 4 ⊢ ∅ ∈ ∪ (𝑅1 “ On)
18		eleq2 2849	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ ∪ (𝑅1 “ On)))
19	17, 18	mpbiri 261	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∅ ∈ 𝑊)
20	4, 14, 19	modelaxrep 46014	. 2 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
21	1, 20	ax-mp 5	1 ⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ∅c0 4278  ∪ cuni 4866  Tr wtr 5211  dom cdm 5650  ran crn 5651   “ cima 5653  Oncon0 6362  Fun wfun 6532  𝑅₁cr1 9766

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-en 8974  df-dom 8975  df-sdom 8976  df-r1 9768  df-rank 9769

This theorem is used by: (None)