Theorem modelaxreplem1 44995

Description: Lemma for modelaxrep 44998. We show that 𝑀 is closed under taking subsets. (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)
modelaxreplem1.5	⊢ 𝐴 ⊆ 𝑥

Assertion

Ref	Expression
modelaxreplem1	⊢ (𝜓 → 𝐴 ∈ 𝑀)

Distinct variable group:   𝑓,𝑀

Allowed substitution hints:   𝜓(𝑥,𝑓)   𝐴(𝑥,𝑓)   𝑀(𝑥)

Proof of Theorem modelaxreplem1

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		modelaxreplem.3	. . 3 ⊢ (𝜓 → ∅ ∈ 𝑀)
2		eleq1 2829	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ∈ 𝑀 ↔ ∅ ∈ 𝑀))
3	1, 2	syl5ibrcom 247	. 2 ⊢ (𝜓 → (𝐴 = ∅ → 𝐴 ∈ 𝑀))
4		vex 3484	. . . . . 6 ⊢ 𝑥 ∈ V
5		modelaxreplem1.5	. . . . . 6 ⊢ 𝐴 ⊆ 𝑥
6	4, 5	ssexi 5322	. . . . 5 ⊢ 𝐴 ∈ V
7	6	0sdom 9147	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
8		ssdomg 9040	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐴 ⊆ 𝑥 → 𝐴 ≼ 𝑥))
9	4, 5, 8	mp2 9	. . . . 5 ⊢ 𝐴 ≼ 𝑥
10		fodomr 9168	. . . . 5 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝑥) → ∃𝑔 𝑔:𝑥–onto→𝐴)
11	9, 10	mpan2 691	. . . 4 ⊢ (∅ ≺ 𝐴 → ∃𝑔 𝑔:𝑥–onto→𝐴)
12	7, 11	sylbir 235	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑔 𝑔:𝑥–onto→𝐴)
13		df-fo 6567	. . . . 5 ⊢ (𝑔:𝑥–onto→𝐴 ↔ (𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴))
14		df-fn 6564	. . . . . . . 8 ⊢ (𝑔 Fn 𝑥 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝑥))
15		modelaxreplem.4	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
16		eleq1 2829	. . . . . . . . . 10 ⊢ (dom 𝑔 = 𝑥 → (dom 𝑔 ∈ 𝑀 ↔ 𝑥 ∈ 𝑀))
17	15, 16	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜓 → (dom 𝑔 = 𝑥 → dom 𝑔 ∈ 𝑀))
18	17	anim2d 612	. . . . . . . 8 ⊢ (𝜓 → ((Fun 𝑔 ∧ dom 𝑔 = 𝑥) → (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀)))
19	14, 18	biimtrid 242	. . . . . . 7 ⊢ (𝜓 → (𝑔 Fn 𝑥 → (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀)))
20		modelaxreplem.1	. . . . . . . . 9 ⊢ (𝜓 → 𝑥 ⊆ 𝑀)
21	5, 20	sstrid 3995	. . . . . . . 8 ⊢ (𝜓 → 𝐴 ⊆ 𝑀)
22		sseq1 4009	. . . . . . . 8 ⊢ (ran 𝑔 = 𝐴 → (ran 𝑔 ⊆ 𝑀 ↔ 𝐴 ⊆ 𝑀))
23	21, 22	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜓 → (ran 𝑔 = 𝐴 → ran 𝑔 ⊆ 𝑀))
24		df-3an 1089	. . . . . . . 8 ⊢ ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀) ∧ ran 𝑔 ⊆ 𝑀))
25		modelaxreplem.2	. . . . . . . . 9 ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
26		funeq 6586	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
27		dmeq 5914	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
28	27	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
29		rneq 5947	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
30	29	sseq1d 4015	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
31	26, 28, 30	3anbi123d 1438	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
32	29	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝑔 ∈ 𝑀))
33	31, 32	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀)))
34	33	spvv 1996	. . . . . . . . 9 ⊢ (∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀) → ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
35	25, 34	syl 17	. . . . . . . 8 ⊢ (𝜓 → ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
36	24, 35	biimtrrid 243	. . . . . . 7 ⊢ (𝜓 → (((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀) ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
37	19, 23, 36	syl2and 608	. . . . . 6 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → ran 𝑔 ∈ 𝑀))
38		eleq1 2829	. . . . . . 7 ⊢ (ran 𝑔 = 𝐴 → (ran 𝑔 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
39	38	adantl 481	. . . . . 6 ⊢ ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → (ran 𝑔 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
40	37, 39	mpbidi 241	. . . . 5 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → 𝐴 ∈ 𝑀))
41	13, 40	biimtrid 242	. . . 4 ⊢ (𝜓 → (𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
42	41	exlimdv 1933	. . 3 ⊢ (𝜓 → (∃𝑔 𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
43	12, 42	syl5 34	. 2 ⊢ (𝜓 → (𝐴 ≠ ∅ → 𝐴 ∈ 𝑀))
44	3, 43	pm2.61dne 3028	1 ⊢ (𝜓 → 𝐴 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  Vcvv 3480   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559   ≼ cdom 8983   ≺ csdm 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986  df-dom 8987  df-sdom 8988

This theorem is referenced by:  modelaxreplem2  44996