Theorem modelaxrep 45438

Description: Conditions which guarantee that a class models the Axiom of Replacement ax-rep 5201. 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 5213). 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 6508	. . . . . 6 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
4		dmeq 5851	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
5	4	eleq1d 2826	. . . . . 6 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
6		rneq 5884	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
7	6	sseq1d 3947	. . . . . 6 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
8	3, 5, 7	3anbi123d 1445	. . . . 5 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
9	6	eleq1d 2826	. . . . 5 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝑔 ∈ 𝑀))
10	8, 9	imbi12d 346	. . . 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 2044	. . 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 220	. 2 ⊢ (𝜓 → ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
13		modelaxrep.3	. 2 ⊢ (𝜓 → ∅ ∈ 𝑀)
14		trss 5191	. . . . . . 7 ⊢ (Tr 𝑀 → (𝑥 ∈ 𝑀 → 𝑥 ⊆ 𝑀))
15	14	imp 408	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑥 ∈ 𝑀) → 𝑥 ⊆ 𝑀)
16	15	ad5ant14 764	. . . . 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 790	. . . . 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 782	. . . . 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 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)) → 𝑥 ∈ 𝑀)
20		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑤(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
21		nfra1 3265	. . . . . 6 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
22	20, 21	nfan 1907	. . . . 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 1922	. . . . . 6 ⊢ Ⅎ𝑧(((Tr 𝑀 ∧ ∀𝑔((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)) ∧ ∅ ∈ 𝑀) ∧ 𝑥 ∈ 𝑀)
24		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑧𝑀
25		nfra1 3265	. . . . . . . 8 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
26	24, 25	nfrexw 3289	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
27	24, 26	nfralw 3288	. . . . . 6 ⊢ Ⅎ𝑧∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
28	23, 27	nfan 1907	. . . . 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 5145	. . . . 5 ⊢ Ⅎ𝑧{⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
30		eqid 2741	. . . . 5 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
31		rsp 3229	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
32	31	adantl 483	. . . . 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 45437	. . . 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 414	. . 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 3133	. 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 844	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 208   ∧ wa 397   ∧ w3a 1093  ∀wal 1546   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ⊆ wss 3884  ∅c0 4263  {copab 5136  Tr wtr 5181  dom cdm 5620  ran crn 5621  Fun wfun 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-en 8888  df-dom 8889  df-sdom 8890

This theorem is referenced by:  wfaxrep  45451