Theorem modelaxreplem2 45428

Description: Lemma for modelaxrep 45430. We define a class 𝐹 and show that the antecedent of Replacement implies that 𝐹 is a function. We use Replacement (in the form of funex 7169) to show that 𝐹 exists. Then we show that, under our hypotheses, the range of 𝐹 is a member of 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)

Hypotheses

Ref	Expression
modelaxreplem.1	⊢ (𝜓 → 𝑥 ⊆ 𝑀)
modelaxreplem.2	⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
modelaxreplem.3	⊢ (𝜓 → ∅ ∈ 𝑀)
modelaxreplem.4	⊢ (𝜓 → 𝑥 ∈ 𝑀)
modelaxreplem2.5	⊢ Ⅎ𝑤𝜓
modelaxreplem2.6	⊢ Ⅎ𝑧𝜓
modelaxreplem2.7	⊢ Ⅎ𝑧𝐹
modelaxreplem2.8	⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
modelaxreplem2.9	⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))

Assertion

Ref	Expression
modelaxreplem2	⊢ (𝜓 → ran 𝐹 ∈ 𝑀)

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

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

Proof of Theorem modelaxreplem2

Step	Hyp	Ref	Expression
1		modelaxreplem2.5	. . . 4 ⊢ Ⅎ𝑤𝜓
2		modelaxreplem.1	. . . . . . 7 ⊢ (𝜓 → 𝑥 ⊆ 𝑀)
3	2	sseld 3921	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
4		modelaxreplem2.9	. . . . . . 7 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
5		nfa1 2157	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
6	5	rmo2i 3827	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃*𝑧 ∈ 𝑀 ∀𝑦𝜑)
7		df-rmo 3343	. . . . . . . 8 ⊢ (∃*𝑧 ∈ 𝑀 ∀𝑦𝜑 ↔ ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))
8	6, 7	sylib 218	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))
9	4, 8	syl6 35	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
10	3, 9	syld 47	. . . . 5 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
11		moanimv 2620	. . . . 5 ⊢ (∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑤 ∈ 𝑥 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
12	10, 11	sylibr 234	. . . 4 ⊢ (𝜓 → ∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
13	1, 12	alrimi 2221	. . 3 ⊢ (𝜓 → ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
14		modelaxreplem2.8	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
15	14	funeqi 6515	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))})
16		funopab 6529	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
17	15, 16	bitri 275	. . 3 ⊢ (Fun 𝐹 ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
18	13, 17	sylibr 234	. 2 ⊢ (𝜓 → Fun 𝐹)
19		modelaxreplem.2	. . 3 ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
20		modelaxreplem.3	. . 3 ⊢ (𝜓 → ∅ ∈ 𝑀)
21		modelaxreplem.4	. . 3 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
22	14	dmeqi 5855	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23		dmopabss 5869	. . . 4 ⊢ dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ⊆ 𝑥
24	22, 23	eqsstri 3969	. . 3 ⊢ dom 𝐹 ⊆ 𝑥
25	2, 19, 20, 21, 24	modelaxreplem1 45427	. 2 ⊢ (𝜓 → dom 𝐹 ∈ 𝑀)
26		an12 646	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
27	26	opabbii 5153	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
28	14, 27	eqtri 2760	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
29	28	rneqi 5888	. . . 4 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
30		rnopabss 5906	. . . 4 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))} ⊆ 𝑀
31	29, 30	eqsstri 3969	. . 3 ⊢ ran 𝐹 ⊆ 𝑀
32	31	a1i 11	. 2 ⊢ (𝜓 → ran 𝐹 ⊆ 𝑀)
33		funex 7169	. . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀) → 𝐹 ∈ V)
34	18, 25, 33	syl2anc 585	. . 3 ⊢ (𝜓 → 𝐹 ∈ V)
35		funeq 6514	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
36		dmeq 5854	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
37	36	eleq1d 2822	. . . . . 6 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝐹 ∈ 𝑀))
38		rneq 5887	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
39	38	sseq1d 3954	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝐹 ⊆ 𝑀))
40	35, 37, 39	3anbi123d 1439	. . . . 5 ⊢ (𝑓 = 𝐹 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀)))
41	38	eleq1d 2822	. . . . 5 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝐹 ∈ 𝑀))
42	40, 41	imbi12d 344	. . . 4 ⊢ (𝑓 = 𝐹 → (((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀) → ran 𝐹 ∈ 𝑀)))
43	42	spcgv 3539	. . 3 ⊢ (𝐹 ∈ V → (∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) → ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀) → ran 𝐹 ∈ 𝑀)))
44	34, 19, 43	sylc 65	. 2 ⊢ (𝜓 → ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀) → ran 𝐹 ∈ 𝑀))
45	18, 25, 32, 44	mp3and 1467	1 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∃*wmo 2538  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3062  ∃*wrmo 3342  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  {copab 5148  dom cdm 5626  ran crn 5627  Fun wfun 6488

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

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 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-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-en 8889  df-dom 8890  df-sdom 8891

This theorem is referenced by:  modelaxreplem3  45429