Theorem onvf1odlem2 35596

Description: Lemma for onvf1od 35599. (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 35595	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
3		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑅1
4		nfrab1 3436	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
5	4	nfint 4922	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
6	3, 5	nffv 6891	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
7		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑣 ∈ 𝐴
8	6, 7	nfrexw 3313	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴
9		eleq1w 2846	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
10	9	notbid 321	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝑣 ∈ 𝐴))
11	10	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘𝑥) ¬ 𝑣 ∈ 𝐴)
12		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (𝑅1‘𝑥) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}))
13	12	rexeqdv 3324	. . . . . . . 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 286	. . . . . . 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 7789	. . . . . 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 6884	. . . . . . 7 ⊢ (𝑅1‘𝑀) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
18	17	rexeqi 3322	. . . . . 6 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴)
19	15, 18	sylibr 237	. . . . 5 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 → ∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴)
20	2, 19	syl 18	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴)
21		df-rex 3090	. . . . 5 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
22		nss 4001	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
23		ssdif0 4321	. . . . . 6 ⊢ ((𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) = ∅)
24	23	necon3bbii 3005	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
25	21, 22, 24	3bitr2i 302	. . . 4 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
26	20, 25	sylib 221	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
27		fvex 6894	. . . . 5 ⊢ (𝑅1‘𝑀) ∈ V
28	27	difexi 5301	. . . 4 ⊢ ((𝑅1‘𝑀) ∖ 𝐴) ∈ V
29		neeq1 3020	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝑧 ≠ ∅ ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅))
30		fveq2 6881	. . . . . 6 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝐺‘𝑧) = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)))
31		id 23	. . . . . 6 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → 𝑧 = ((𝑅1‘𝑀) ∖ 𝐴))
32	30, 31	eleq12d 2857	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝐺‘𝑧) ∈ 𝑧 ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
33	29, 32	imbi12d 347	. . . 4 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧) ↔ (((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅ → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))))
34	28, 33	spcv 3564	. . 3 ⊢ (∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧) → (((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅ → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
35	1, 26, 34	syl2im 41	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
36		onvf1odlem2.3	. . 3 ⊢ 𝑁 = (𝐺‘((𝑅1‘𝑀) ∖ 𝐴))
37	36	eleq1i 2854	. 2 ⊢ (𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴) ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))
38	35, 37	imbitrrdi 255	1 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  {crab 3416   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ∩ cint 4912  Oncon0 6360  ‘cfv 6536  𝑅₁cr1 9730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-r1 9732

This theorem is used by:  onvf1odlem4  35598  onvf1od  35599