Theorem modelaxrep 45300

Description: Conditions which guarantee that a class models the Axiom of Replacement ax-rep 5225. Similar to Lemma II.2.4(6) of [Kunen2] p. 111. The first two hypotheses are those in Kunen. The reason for the third hypothesis that our version of Replacement is different from Kunen's (which is zfrep6 7902). If we assumed Regularity, we could eliminate this extra hypothesis, since under Regularity, the empty set is a member of every non-empty transitive class.

Note that, to obtain the relativization of an instance of Replacement to 𝑀, the formula ∀𝑦𝜑 would need to be replaced with ∀𝑦 ∈ 𝑀𝜒, where 𝜒 is 𝜑 with all quantifiers relativized to 𝑀. However, we can obtain this by using 𝑦 ∈ 𝑀 ∧ 𝜒 for 𝜑 in this theorem, so it does establish that all instances of Replacement hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)

Hypotheses

Ref	Expression
modelaxrep.1	⊢ (𝜓 → Tr 𝑀)
modelaxrep.2	⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
modelaxrep.3	⊢ (𝜓 → ∅ ∈ 𝑀)

Assertion

Ref	Expression
modelaxrep	⊢ (𝜓 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑀   𝑓,𝑀

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

Proof of Theorem modelaxrep

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		modelaxrep.1	. 2 ⊢ (𝜓 → Tr 𝑀)
2		modelaxrep.2	. . 3 ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
3		funeq 6513	. . . . . 6 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
4		dmeq 5853	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
5	4	eleq1d 2822	. . . . . 6 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
6		rneq 5886	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
7	6	sseq1d 3966	. . . . . 6 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
8	3, 5, 7	3anbi123d 1439	. . . . 5 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
9	6	eleq1d 2822	. . . . 5 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝑔 ∈ 𝑀))
10	8, 9	imbi12d 344	. . . 4 ⊢ (𝑓 = 𝑔 → (((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)))
11	10	cbvalvw 2038	. . 3 ⊢ (∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
12	2, 11	sylib 218	. 2 ⊢ (𝜓 → ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
13		modelaxrep.3	. 2 ⊢ (𝜓 → ∅ ∈ 𝑀)
14		trss 5216	. . . . . . 7 ⊢ (Tr 𝑀 → (𝑥 ∈ 𝑀 → 𝑥 ⊆ 𝑀))
15	14	imp 406	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → 𝑥 ⊆ 𝑀)
16	15	ad5ant14 758	. . . . 5 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → 𝑥 ⊆ 𝑀)
17		simp-4r 784	. . . . 5 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
18		simpllr 776	. . . . 5 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∅ ∈ 𝑀)
19		simplr 769	. . . . 5 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → 𝑥 ∈ 𝑀)
20		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑤(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
21		nfra1 3261	. . . . . 6 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
22	20, 21	nfan 1901	. . . . 5 ⊢ Ⅎ𝑤((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦))
23		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
24		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑧𝑀
25		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
26	24, 25	nfrexw 3285	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
27	24, 26	nfralw 3284	. . . . . 6 ⊢ Ⅎ𝑧∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
28	23, 27	nfan 1901	. . . . 5 ⊢ Ⅎ𝑧((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦))
29		nfopab2 5170	. . . . 5 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
30		eqid 2737	. . . . 5 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
31		rsp 3225	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
32	31	adantl 481	. . . . 5 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
33	16, 17, 18, 19, 22, 28, 29, 30, 32	modelaxreplem3 45299	. . . 4 ⊢ (((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) ∧ ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
34	33	ex 412	. . 3 ⊢ ((((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀) → (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
35	34	ralrimiva 3129	. 2 ⊢ (((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
36	1, 12, 13, 35	syl21anc 838	1 ⊢ (𝜓 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902  ∅c0 4286  {copab 5161  Tr wtr 5206  dom cdm 5625  ran crn 5626  Fun wfun 6487

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-en 8889  df-dom 8890  df-sdom 8891

This theorem is referenced by:  wfaxrep  45313