Theorem modelaxreplem2 45716

Description: Lemma for modelaxrep 45718. 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 7217) 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 3936	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
4		modelaxreplem2.9	. . . . . . 7 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
5		nfa1 2186	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
6	5	rmo2i 3841	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃*𝑧 ∈ 𝑀 ∀𝑦𝜑)
7		df-rmo 3369	. . . . . . . 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 2647	. . . . 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 2249	. . 3 ⊢ (𝜓 → ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
14		modelaxreplem2.8	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
15	14	funeqi 6557	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))})
16		funopab 6571	. . . 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 5894	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23		dmopabss 5908	. . . 4 ⊢ dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ⊆ 𝑥
24	22, 23	eqsstri 3983	. . 3 ⊢ dom 𝐹 ⊆ 𝑥
25	2, 19, 20, 21, 24	modelaxreplem1 45715	. 2 ⊢ (𝜓 → dom 𝐹 ∈ 𝑀)
26		an12 657	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
27	26	opabbii 5178	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
28	14, 27	eqtri 2786	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
29	28	rneqi 5927	. . . 4 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
30		rnopabss 5945	. . . 4 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))} ⊆ 𝑀
31	29, 30	eqsstri 3983	. . 3 ⊢ ran 𝐹 ⊆ 𝑀
32	31	a1i 11	. 2 ⊢ (𝜓 → ran 𝐹 ⊆ 𝑀)
33		funex 7217	. . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀) → 𝐹 ∈ V)
34	18, 25, 33	syl2anc 595	. . 3 ⊢ (𝜓 → 𝐹 ∈ V)
35		funeq 6556	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
36		dmeq 5893	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
37	36	eleq1d 2848	. . . . . 6 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝐹 ∈ 𝑀))
38		rneq 5926	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
39	38	sseq1d 3968	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝐹 ⊆ 𝑀))
40	35, 37, 39	3anbi123d 1464	. . . . 5 ⊢ (𝑓 = 𝐹 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀)))
41	38	eleq1d 2848	. . . . 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 3555	. . 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 1493	1 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  ∃*wmo 2565  Ⅎwnfc 2910  ∀wral 3079  ∃wrex 3089  ∃*wrmo 3368  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  {copab 5173  dom cdm 5661  ran crn 5662  Fun wfun 6530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-en 8940  df-dom 8941  df-sdom 8942

This theorem is used by:  modelaxreplem3  45717