Theorem modelaxreplem1 45788

Description: Lemma for modelaxrep 45791. 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 2850	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ∈ 𝑀 ↔ ∅ ∈ 𝑀))
3	1, 2	syl5ibrcom 250	. 2 ⊢ (𝜓 → (𝐴 = ∅ → 𝐴 ∈ 𝑀))
4		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
5		modelaxreplem1.5	. . . . . 6 ⊢ 𝐴 ⊆ 𝑥
6	4, 5	ssexi 5291	. . . . 5 ⊢ 𝐴 ∈ V
7	6	0sdom 9109	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
8		ssdomg 9009	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐴 ⊆ 𝑥 → 𝐴 ≼ 𝑥))
9	4, 5, 8	mp2 9	. . . . 5 ⊢ 𝐴 ≼ 𝑥
10		fodomr 9129	. . . . 5 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝑥) → ∃𝑔 𝑔:𝑥–onto→𝐴)
11	9, 10	mpan2 704	. . . 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 2850	. . . . . . . . . 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 624	. . . . . . . 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 3945	. . . . . . . 8 ⊢ (𝜓 → 𝐴 ⊆ 𝑀)
22		sseq1 3959	. . . . . . . 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 1105	. . . . . . . 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 5891	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
28	27	eleq1d 2847	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
29		rneq 5924	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
30	29	sseq1d 3965	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
31	26, 28, 30	3anbi123d 1464	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
32	29	eleq1d 2847	. . . . . . . . . . 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 2021	. . . . . . . . 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 620	. . . . . 6 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → ran 𝑔 ∈ 𝑀))
38		eleq1 2850	. . . . . . 7 ⊢ (ran 𝑔 = 𝐴 → (ran 𝑔 ∈ 𝑀 ↔ 𝐴 ∈ 𝑀))
39	38	adantl 487	. . . . . 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 1966	. . 3 ⊢ (𝜓 → (∃𝑔 𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
43	12, 42	syl5 35	. 2 ⊢ (𝜓 → (𝐴 ≠ ∅ → 𝐴 ∈ 𝑀))
44	3, 43	pm2.61dne 3043	1 ⊢ (𝜓 → 𝐴 ∈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  dom cdm 5659  ran crn 5660  Fun wfun 6531   Fn wfn 6532  –onto→wfo 6535   ≼ cdom 8953   ≺ csdm 8954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-en 8956  df-dom 8957  df-sdom 8958

This theorem is used by:  modelaxreplem2  45789