Theorem wfaxrep 44984

Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5214. 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 44949	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
3		treq 5202	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → Tr 𝑊)
5		vex 3437	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6	5	rnex 7834	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
7	6	r1elss 9690	. . . . . . . 8 ⊢ (ran 𝑓 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On))
8	7	biimpri 228	. . . . . . 7 ⊢ (ran 𝑓 ⊆ ∪ (𝑅1 “ On) → ran 𝑓 ∈ ∪ (𝑅1 “ On))
9	1	sseq2i 3961	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝑊 ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On))
10	1	eleq2i 2820	. . . . . . 7 ⊢ (ran 𝑓 ∈ 𝑊 ↔ ran 𝑓 ∈ ∪ (𝑅1 “ On))
11	8, 9, 10	3imtr4i 292	. . . . . 6 ⊢ (ran 𝑓 ⊆ 𝑊 → ran 𝑓 ∈ 𝑊)
12	11	3ad2ant3 1135	. . . . 5 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊)
13	12	ax-gen 1795	. . . 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 9714	. . . . 5 ⊢ On ⊆ ∪ (𝑅1 “ On)
16		0elon 6356	. . . . 5 ⊢ ∅ ∈ On
17	15, 16	sselii 3928	. . . 4 ⊢ ∅ ∈ ∪ (𝑅1 “ On)
18		eleq2 2817	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ ∪ (𝑅1 “ On)))
19	17, 18	mpbiri 258	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∅ ∈ 𝑊)
20	4, 14, 19	modelaxrep 44971	. 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3899  ∅c0 4280  ∪ cuni 4856  Tr wtr 5195  dom cdm 5613  ran crn 5614   “ cima 5616  Oncon0 6301  Fun wfun 6470  𝑅₁cr1 9646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4895  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7343  df-om 7791  df-2nd 7916  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-en 8864  df-dom 8865  df-sdom 8866  df-r1 9648  df-rank 9649

This theorem is referenced by: (None)