Theorem modelaxreplem1 45405

Description: Lemma for modelaxrep 45408. 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 2824	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ∈ 𝑀 ↔ ∅ ∈ 𝑀))
3	1, 2	syl5ibrcom 247	. 2 ⊢ (𝜓 → (𝐴 = ∅ → 𝐴 ∈ 𝑀))
4		vex 3433	. . . . . 6 ⊢ 𝑥 ∈ V
5		modelaxreplem1.5	. . . . . 6 ⊢ 𝐴 ⊆ 𝑥
6	4, 5	ssexi 5263	. . . . 5 ⊢ 𝐴 ∈ V
7	6	0sdom 9046	. . . 4 ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
8		ssdomg 8947	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐴 ⊆ 𝑥 → 𝐴 ≼ 𝑥))
9	4, 5, 8	mp2 9	. . . . 5 ⊢ 𝐴 ≼ 𝑥
10		fodomr 9066	. . . . 5 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝑥) → ∃𝑔 𝑔:𝑥–onto→𝐴)
11	9, 10	mpan2 692	. . . 4 ⊢ (∅ ≺ 𝐴 → ∃𝑔 𝑔:𝑥–onto→𝐴)
12	7, 11	sylbir 235	. . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑔 𝑔:𝑥–onto→𝐴)
13		df-fo 6504	. . . . 5 ⊢ (𝑔:𝑥–onto→𝐴 ↔ (𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴))
14		df-fn 6501	. . . . . . . 8 ⊢ (𝑔 Fn 𝑥 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝑥))
15		modelaxreplem.4	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
16		eleq1 2824	. . . . . . . . . 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 613	. . . . . . . 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 3933	. . . . . . . 8 ⊢ (𝜓 → 𝐴 ⊆ 𝑀)
22		sseq1 3947	. . . . . . . 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 6518	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
27		dmeq 5858	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
28	27	eleq1d 2821	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (dom 𝑓 ∈ 𝑀 ↔ dom 𝑔 ∈ 𝑀))
29		rneq 5891	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
30	29	sseq1d 3953	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (ran 𝑓 ⊆ 𝑀 ↔ ran 𝑔 ⊆ 𝑀))
31	26, 28, 30	3anbi123d 1439	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔 ∈ 𝑀 ∧ ran 𝑔 ⊆ 𝑀)))
32	29	eleq1d 2821	. . . . . . . . . . 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 1990	. . . . . . . . 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 609	. . . . . 6 ⊢ (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → ran 𝑔 ∈ 𝑀))
38		eleq1 2824	. . . . . . 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 1935	. . 3 ⊢ (𝜓 → (∃𝑔 𝑔:𝑥–onto→𝐴 → 𝐴 ∈ 𝑀))
43	12, 42	syl5 34	. 2 ⊢ (𝜓 → (𝐴 ≠ ∅ → 𝐴 ∈ 𝑀))
44	3, 43	pm2.61dne 3018	1 ⊢ (𝜓 → 𝐴 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085  dom cdm 5631  ran crn 5632  Fun wfun 6492   Fn wfn 6493  –onto→wfo 6496   ≼ cdom 8891   ≺ csdm 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-en 8894  df-dom 8895  df-sdom 8896

This theorem is referenced by:  modelaxreplem2  45406