Theorem wfaxrep 45424

Description: The class of well-founded sets models the Axiom of Replacement ax-rep 5212. 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 45389	. . . 4 ⊢ Tr ∪ (𝑅1 “ On)
3		treq 5200	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → Tr 𝑊)
5		vex 3434	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6	5	rnex 7852	. . . . . . . . 9 ⊢ ran 𝑓 ∈ V
7	6	r1elss 9719	. . . . . . . 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 3952	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝑊 ↔ ran 𝑓 ⊆ ∪ (𝑅1 “ On))
10	1	eleq2i 2829	. . . . . . 7 ⊢ (ran 𝑓 ∈ 𝑊 ↔ ran 𝑓 ∈ ∪ (𝑅1 “ On))
11	8, 9, 10	3imtr4i 292	. . . . . 6 ⊢ (ran 𝑓 ⊆ 𝑊 → ran 𝑓 ∈ 𝑊)
12	11	3ad2ant3 1136	. . . . 5 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑊 ∧ ran 𝑓 ⊆ 𝑊) → ran 𝑓 ∈ 𝑊)
13	12	ax-gen 1797	. . . 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 9743	. . . . 5 ⊢ On ⊆ ∪ (𝑅1 “ On)
16		0elon 6370	. . . . 5 ⊢ ∅ ∈ On
17	15, 16	sselii 3919	. . . 4 ⊢ ∅ ∈ ∪ (𝑅1 “ On)
18		eleq2 2826	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (∅ ∈ 𝑊 ↔ ∅ ∈ ∪ (𝑅1 “ On)))
19	17, 18	mpbiri 258	. . 3 ⊢ (𝑊 = ∪ (𝑅1 “ On) → ∅ ∈ 𝑊)
20	4, 14, 19	modelaxrep 45411	. 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 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  ∪ cuni 4851  Tr wtr 5193  dom cdm 5622  ran crn 5623   “ cima 5625  Oncon0 6315  Fun wfun 6484  𝑅₁cr1 9675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-en 8885  df-dom 8886  df-sdom 8887  df-r1 9677  df-rank 9678

This theorem is referenced by: (None)