Theorem modelaxreplem1 45613

Description: Lemma for modelaxrep 45616. 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 2857	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ∈ 𝑀 ↔ ∅ ∈ 𝑀))
3	1, 2	syl5ibrcom 250	. 2 ⊢ (𝜓 → (𝐴 = ∅ → 𝐴 ∈ 𝑀))
4		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
5		modelaxreplem1.5	. . . . . 6 ⊢ 𝐴 ⊆ 𝑥
6	4, 5	ssexi 5293	. . . . 5 ⊢ 𝐴 ∈ V
7	6	0sdom 9096	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
8		ssdomg 8997	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐴 ⊆ 𝑥 → 𝐴 ≼ 𝑥))
9	4, 5, 8	mp2 9	. . . . 5 ⊢ 𝐴 ≼ 𝑥
10		fodomr 9116	. . . . 5 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝑥) → ∃𝑔 𝑔:𝑥–onto→𝐴)
11	9, 10	mpan2 703	. . . 4 ⊢ (∅ ≺ 𝐴 → ∃𝑔 𝑔:𝑥–onto→𝐴)
12	7, 11	sylbir 238	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑔 𝑔:𝑥–onto→𝐴)
13		df-fo 6543	. . . . 5 ⊢ (𝑔:𝑥–onto→𝐴 ↔ (𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴))
14		df-fn 6540	. . . . . . . 8 ⊢ (𝑔 Fn 𝑥 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝑥))
15		modelaxreplem.4	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
16		eleq1 2857	. . . . . . . . . 10 ⊢ (dom 𝑔 = 𝑥 → (dom 𝑔 ∈ 𝑀 ↔ 𝑥 ∈ 𝑀))
17	15, 16	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝜓 → (dom 𝑔 = 𝑥 → dom 𝑔 ∈ 𝑀))
18	17	anim2d 623	. . . . . . . 8 ⊢ (𝜓 → ((Fun 𝑔 ∧ dom 𝑔 = 𝑥) → (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀)))
19	14, 18	biimtrid 245	. . . . . . 7 ⊢ (𝜓 → (𝑔 Fn 𝑥 → (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀)))
20		modelaxreplem.1	. . . . . . . . 9 ⊢ (𝜓 → 𝑥 ⊆ 𝑀)
21	5, 20	sstrid 3956	. . . . . . . 8 ⊢ (𝜓 → 𝐴 ⊆ 𝑀)
22		sseq1 3970	. . . . . . . 8 ⊢ (ran 𝑔 = 𝐴 → (ran 𝑔 ⊆ 𝑀 ↔ 𝐴 ⊆ 𝑀))
23	21, 22	syl5ibrcom 250	. . . . . . 7 ⊢ (𝜓 → (ran 𝑔 = 𝐴 → ran 𝑔 ⊆ 𝑀))
24		df-3an 1103	. . . . . . . 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 6557	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
27		dmeq 5894	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
28	27	eleq1d 2854	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
29		rneq 5927	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
30	29	sseq1d 3976	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
31	26, 28, 30	3anbi123d 1462	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
32	29	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ∈ 𝑀 ↔ ran 𝑔 ∈ 𝑀))
33	31, 32	imbi12d 347	. . . . . . . . . 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 2015	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜓 → ((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
36	24, 35	biimtrrid 246	. . . . . . 7 ⊢ (𝜓 → (((Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀) ∧ ran 𝑔 ⊆ 𝑀) → ran 𝑔 ∈ 𝑀))
37	19, 23, 36	syl2and 619	. . . . . 6 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → ran 𝑔 ∈ 𝑀))
38		eleq1 2857	. . . . . . 7 ⊢ (ran 𝑔 = 𝐴 → (ran 𝑔 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
39	38	adantl 486	. . . . . 6 ⊢ ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → (ran 𝑔 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
40	37, 39	mpbidi 244	. . . . 5 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → 𝐴 ∈ 𝑀))
41	13, 40	biimtrid 245	. . . 4 ⊢ (𝜓 → (𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
42	41	exlimdv 1960	. . 3 ⊢ (𝜓 → (∃𝑔 𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
43	12, 42	syl5 35	. 2 ⊢ (𝜓 → (𝐴 ≠ ∅ → 𝐴 ∈ 𝑀))
44	3, 43	pm2.61dne 3050	1 ⊢ (𝜓 → 𝐴 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113  dom cdm 5662  ran crn 5663  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535   ≼ cdom 8941   ≺ csdm 8942

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-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-rab 3424  df-v 3465  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-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-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8944  df-dom 8945  df-sdom 8946

This theorem is referenced by:  modelaxreplem2  45614