Theorem modelaxreplem1 44962

Description: Lemma for modelaxrep 44965. 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 2816	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ∈ 𝑀 ↔ ∅ ∈ 𝑀))
3	1, 2	syl5ibrcom 247	. 2 ⊢ (𝜓 → (𝐴 = ∅ → 𝐴 ∈ 𝑀))
4		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
5		modelaxreplem1.5	. . . . . 6 ⊢ 𝐴 ⊆ 𝑥
6	4, 5	ssexi 5261	. . . . 5 ⊢ 𝐴 ∈ V
7	6	0sdom 9025	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
8		ssdomg 8925	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐴 ⊆ 𝑥 → 𝐴 ≼ 𝑥))
9	4, 5, 8	mp2 9	. . . . 5 ⊢ 𝐴 ≼ 𝑥
10		fodomr 9045	. . . . 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 6488	. . . . 5 ⊢ (𝑔:𝑥–onto→𝐴 ↔ (𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴))
14		df-fn 6485	. . . . . . . 8 ⊢ (𝑔 Fn 𝑥 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝑥))
15		modelaxreplem.4	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
16		eleq1 2816	. . . . . . . . . 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 3947	. . . . . . . 8 ⊢ (𝜓 → 𝐴 ⊆ 𝑀)
22		sseq1 3961	. . . . . . . 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 1088	. . . . . . . 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 6502	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
27		dmeq 5846	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
28	27	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
29		rneq 5878	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
30	29	sseq1d 3967	. . . . . . . . . . . 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 2813	. . . . . . . . . . 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 1988	. . . . . . . . 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 2816	. . . . . . 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 3011	1 ⊢ (𝜓 → 𝐴 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092  dom cdm 5619  ran crn 5620  Fun wfun 6476   Fn wfn 6477  –onto→wfo 6480   ≼ cdom 8870   ≺ csdm 8871

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-en 8873  df-dom 8874  df-sdom 8875

This theorem is referenced by:  modelaxreplem2  44963