Theorem modelaxreplem2 45560

Description: Lemma for modelaxrep 45562. 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 7205) 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 3937	. . . . . 6 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
4		modelaxreplem2.9	. . . . . . 7 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
5		nfa1 2187	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
6	5	rmo2i 3843	. . . . . . . 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 220	. . . . . . 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 2648	. . . . 5 ⊢ (∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑤 ∈ 𝑥 → ∃*𝑧(𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
12	10, 11	sylibr 236	. . . 4 ⊢ (𝜓 → ∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
13	1, 12	alrimi 2250	. . 3 ⊢ (𝜓 → ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
14		modelaxreplem2.8	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
15	14	funeqi 6544	. . . 4 ⊢ (Fun 𝐹 ↔ Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))})
16		funopab 6558	. . . 4 ⊢ (Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
17	15, 16	bitri 277	. . 3 ⊢ (Fun 𝐹 ↔ ∀𝑤∃*𝑧(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
18	13, 17	sylibr 236	. 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 5882	. . . 4 ⊢ dom 𝐹 = dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23		dmopabss 5896	. . . 4 ⊢ dom {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} ⊆ 𝑥
24	22, 23	eqsstri 3984	. . 3 ⊢ dom 𝐹 ⊆ 𝑥
25	2, 19, 20, 21, 24	modelaxreplem1 45559	. 2 ⊢ (𝜓 → dom 𝐹 ∈ 𝑀)
26		an12 655	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
27	26	opabbii 5169	. . . . . 6 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
28	14, 27	eqtri 2787	. . . . 5 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
29	28	rneqi 5915	. . . 4 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))}
30		rnopabss 5933	. . . 4 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))} ⊆ 𝑀
31	29, 30	eqsstri 3984	. . 3 ⊢ ran 𝐹 ⊆ 𝑀
32	31	a1i 11	. 2 ⊢ (𝜓 → ran 𝐹 ⊆ 𝑀)
33		funex 7205	. . . 4 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀) → 𝐹 ∈ V)
34	18, 25, 33	syl2anc 593	. . 3 ⊢ (𝜓 → 𝐹 ∈ V)
35		funeq 6543	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
36		dmeq 5881	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
37	36	eleq1d 2849	. . . . . 6 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝐹 ∈ 𝑀))
38		rneq 5914	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
39	38	sseq1d 3969	. . . . . 6 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝐹 ⊆ 𝑀))
40	35, 37, 39	3anbi123d 1459	. . . . 5 ⊢ (𝑓 = 𝐹 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝐹 ∧ dom 𝐹 ∈ 𝑀 ∧ ran 𝐹 ⊆ 𝑀)))
41	38	eleq1d 2849	. . . . 5 ⊢ (𝑓 = 𝐹 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝐹 ∈ 𝑀))
42	40, 41	imbi12d 346	. . . 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 3557	. . 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 1487	1 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099  ∀wal 1560   = wceq 1562  Ⅎwnf 1805   ∈ wcel 2144  ∃*wmo 2566  Ⅎwnfc 2911  ∀wral 3078  ∃wrex 3088  ∃*wrmo 3368  Vcvv 3456   ⊆ wss 3906  ∅c0 4287  {copab 5164  dom cdm 5649  ran crn 5650  Fun wfun 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-en 8930  df-dom 8931  df-sdom 8932

This theorem is referenced by:  modelaxreplem3  45561