Theorem onvf1odlem2 35247

Description: Lemma for onvf1od 35250. (Contributed by BTernaryTau, 2-Dec-2025.)

Hypotheses

Ref	Expression
onvf1odlem2.1	⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
onvf1odlem2.2	⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
onvf1odlem2.3	⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴))

Assertion

Ref	Expression
onvf1odlem2	⊢ (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐺   𝑧,𝑀   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem onvf1odlem2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onvf1odlem2.1	. . 3 ⊢ (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))
2		onvf1odlem1 35246	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
3		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑅1
4		nfrab1 3417	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
5	4	nfint 4910	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
6	3, 5	nffv 6842	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
7		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑣 ∈ 𝐴
8	6, 7	nfrexw 3282	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴
9		eleq1w 2817	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
10	9	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝑣 ∈ 𝐴))
11	10	cbvrexvw 3213	. . . . . . . 8 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘𝑥) ¬ 𝑣 ∈ 𝐴)
12		fveq2 6832	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (𝑅1‘𝑥) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}))
13	12	rexeqdv 3295	. . . . . . . 8 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (∃𝑣 ∈ (𝑅1‘𝑥) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴))
14	11, 13	bitrid 283	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴))
15	8, 14	onminsb 7737	. . . . . 6 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 → ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴)
16		onvf1odlem2.2	. . . . . . . 8 ⊢ 𝑀 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
17	16	fveq2i 6835	. . . . . . 7 ⊢ (𝑅1‘𝑀) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
18	17	rexeqi 3293	. . . . . 6 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴)
19	15, 18	sylibr 234	. . . . 5 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 → ∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴)
20	2, 19	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴)
21		df-rex 3059	. . . . 5 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
22		nss 3996	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
23		ssdif0 4316	. . . . . 6 ⊢ ((𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) = ∅)
24	23	necon3bbii 2977	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
25	21, 22, 24	3bitr2i 299	. . . 4 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
26	20, 25	sylib 218	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
27		fvex 6845	. . . . 5 ⊢ (𝑅1‘𝑀) ∈ V
28	27	difexi 5273	. . . 4 ⊢ ((𝑅1‘𝑀) ∖ 𝐴) ∈ V
29		neeq1 2992	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝑧 ≠ ∅ ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅))
30		fveq2 6832	. . . . . 6 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝐺‘𝑧) = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)))
31		id 22	. . . . . 6 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → 𝑧 = ((𝑅1‘𝑀) ∖ 𝐴))
32	30, 31	eleq12d 2828	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝐺‘𝑧) ∈ 𝑧 ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
33	29, 32	imbi12d 344	. . . 4 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧) ↔ (((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅ → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))))
34	28, 33	spcv 3557	. . 3 ⊢ (∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧) → (((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅ → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
35	1, 26, 34	syl2im 40	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
36		onvf1odlem2.3	. . 3 ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴))
37	36	eleq1i 2825	. 2 ⊢ (𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴) ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))
38	35, 37	imbitrrdi 252	1 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283  ∩ cint 4900  Oncon0 6315  ‘cfv 6490  𝑅₁cr1 9672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-r1 9674

This theorem is referenced by:  onvf1odlem4  35249  onvf1od  35250