Theorem modelaxrep 45396

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 6507	. . . . . 6 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
4		dmeq 5847	. . . . . . 7 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
5	4	eleq1d 2820	. . . . . 6 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
6		rneq 5880	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
7	6	sseq1d 3948	. . . . . 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 2820	. . . . 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 5191	. . . . . . 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 3259	. . . . . 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 2897	. . . . . . 7 ⊢ Ⅎ𝑧𝑀
25		nfra1 3259	. . . . . . . 8 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
26	24, 25	nfrexw 3283	. . . . . . 7 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)
27	24, 26	nfralw 3282	. . . . . 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 5145	. . . . 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 3223	. . . . . 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 45395	. . . 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 3127	. 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 3049  ∃wrex 3059   ⊆ wss 3885  ∅c0 4263  {copab 5136  Tr wtr 5181  dom cdm 5620  ran crn 5621  Fun wfun 6481

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  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 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-en 8883  df-dom 8884  df-sdom 8885

This theorem is referenced by:  wfaxrep  45409