Theorem modelaxreplem2 45614

Description: Lemma for modelaxrep 45616. 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 7218) 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 3944	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
4		modelaxreplem2.9	. . . . . . 7 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
5		nfa1 2192	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
6	5	rmo2i 3850	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃*𝑧 ∈ 𝑀 ∀𝑦𝜑)
7		df-rmo 3376	. . . . . . . 8 ⊢ (∃*𝑧 ∈ 𝑀 ∀𝑦𝜑 ↔ ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))
8	6, 7	sylib 221	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))
9	4, 8	syl6 36	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
10	3, 9	syld 48	. . . . 5 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
11		moanimv 2653	. . . . 5 ⊢ (∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑤 ∈ 𝑥 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
12	10, 11	sylibr 237	. . . 4 ⊢ (𝜓 → ∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
13	1, 12	alrimi 2255	. . 3 ⊢ (𝜓 → ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
14		modelaxreplem2.8	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
15	14	funeqi 6558	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))})
16		funopab 6572	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
17	15, 16	bitri 278	. . 3 ⊢ (Fun 𝐹 ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
18	13, 17	sylibr 237	. 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 5895	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23		dmopabss 5909	. . . 4 ⊢ dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ⊆ 𝑥
24	22, 23	eqsstri 3991	. . 3 ⊢ dom 𝐹 ⊆ 𝑥
25	2, 19, 20, 21, 24	modelaxreplem1 45613	. 2 ⊢ (𝜓 → dom 𝐹 ∈ 𝑀)
26		an12 657	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
27	26	opabbii 5182	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
28	14, 27	eqtri 2792	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
29	28	rneqi 5928	. . . 4 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
30		rnopabss 5946	. . . 4 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))} ⊆ 𝑀
31	29, 30	eqsstri 3991	. . 3 ⊢ ran 𝐹 ⊆ 𝑀
32	31	a1i 11	. 2 ⊢ (𝜓 → ran 𝐹 ⊆ 𝑀)
33		funex 7218	. . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀) → 𝐹 ∈ V)
34	18, 25, 33	syl2anc 595	. . 3 ⊢ (𝜓 → 𝐹 ∈ V)
35		funeq 6557	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
36		dmeq 5894	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
37	36	eleq1d 2854	. . . . . 6 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝐹 ∈ 𝑀))
38		rneq 5927	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
39	38	sseq1d 3976	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝐹 ⊆ 𝑀))
40	35, 37, 39	3anbi123d 1462	. . . . 5 ⊢ (𝑓 = 𝐹 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀)))
41	38	eleq1d 2854	. . . . 5 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝐹 ∈ 𝑀))
42	40, 41	imbi12d 347	. . . 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 3564	. . 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 66	. 2 ⊢ (𝜓 → ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀) → ran 𝐹 ∈ 𝑀))
45	18, 25, 32, 44	mp3and 1490	1 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  ∃*wmo 2571  Ⅎwnfc 2916  ∀wral 3085  ∃wrex 3095  ∃*wrmo 3375  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  {copab 5177  dom cdm 5662  ran crn 5663  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-en 8944  df-dom 8945  df-sdom 8946

This theorem is referenced by:  modelaxreplem3  45615