Theorem modelaxrep 44954

Description: Conditions which guarantee that a class models the Axiom of Replacement ax-rep 5249. 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 7951). 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 6555	. . . . . 6 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
4		dmeq 5883	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
5	4	eleq1d 2819	. . . . . 6 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
6		rneq 5916	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
7	6	sseq1d 3990	. . . . . 6 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
8	3, 5, 7	3anbi123d 1438	. . . . 5 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
9	6	eleq1d 2819	. . . . 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 2035	. . 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 5240	. . . . . . 7 ⊢ (Tr 𝑀 → (𝑥 ∈ 𝑀 → 𝑥 ⊆ 𝑀))
15	14	imp 406	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → 𝑥 ⊆ 𝑀)
16	15	ad5ant14 757	. . . . 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 783	. . . . 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 775	. . . . 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 768	. . . . 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 1914	. . . . . 6 ⊢ Ⅎ𝑤(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
21		nfra1 3266	. . . . . 6 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
22	20, 21	nfan 1899	. . . . 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 1914	. . . . . 6 ⊢ Ⅎ𝑧(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
24		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑧𝑀
25		nfra1 3266	. . . . . . . 8 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
26	24, 25	nfrexw 3293	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
27	24, 26	nfralw 3291	. . . . . 6 ⊢ Ⅎ𝑧∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
28	23, 27	nfan 1899	. . . . 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 5190	. . . . 5 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
30		eqid 2735	. . . . 5 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
31		rsp 3230	. . . . . 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 44953	. . . 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 3132	. 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 837	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 1086  ∀wal 1538   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∅c0 4308  {copab 5181  Tr wtr 5229  dom cdm 5654  ran crn 5655  Fun wfun 6524

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-en 8958  df-dom 8959  df-sdom 8960

This theorem is referenced by:  wfaxrep  44967