Theorem onvf1odlem2 35451

Description: Lemma for onvf1od 35454. (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 35450	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
3		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑅1
4		nfrab1 3436	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
5	4	nfint 4917	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
6	3, 5	nffv 6879	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
7		nfv 1936	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑣 ∈ 𝐴
8	6, 7	nfrexw 3312	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴
9		eleq1w 2847	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
10	9	notbid 320	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝑣 ∈ 𝐴))
11	10	cbvrexvw 3243	. . . . . . . 8 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘𝑥) ¬ 𝑣 ∈ 𝐴)
12		fveq2 6869	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (𝑅1‘𝑥) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}))
13	12	rexeqdv 3323	. . . . . . . 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 285	. . . . . . 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 7779	. . . . . 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 6872	. . . . . . 7 ⊢ (𝑅1‘𝑀) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
18	17	rexeqi 3321	. . . . . 6 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴)
19	15, 18	sylibr 236	. . . . 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 3089	. . . . 5 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
22		nss 4002	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
23		ssdif0 4321	. . . . . 6 ⊢ ((𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) = ∅)
24	23	necon3bbii 3006	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
25	21, 22, 24	3bitr2i 301	. . . 4 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
26	20, 25	sylib 220	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅)
27		fvex 6882	. . . . 5 ⊢ (𝑅1‘𝑀) ∈ V
28	27	difexi 5288	. . . 4 ⊢ ((𝑅1‘𝑀) ∖ 𝐴) ∈ V
29		neeq1 3021	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝑧 ≠ ∅ ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅))
30		fveq2 6869	. . . . . 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 2858	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝐺‘𝑧) ∈ 𝑧 ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴)))
33	29, 32	imbi12d 346	. . . 4 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → ((𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧) ↔ (((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅ → (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))))
34	28, 33	spcv 3566	. . 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 2855	. 2 ⊢ (𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴) ↔ (𝐺‘((𝑅1‘𝑀) ∖ 𝐴)) ∈ ((𝑅1‘𝑀) ∖ 𝐴))
38	35, 37	imbitrrdi 254	1 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → 𝑁 ∈ ((𝑅1‘𝑀) ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  {crab 3416   ∖ cdif 3903   ⊆ wss 3906  ∅c0 4287  ∩ cint 4907  Oncon0 6348  ‘cfv 6523  𝑅₁cr1 9722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-om 7849  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-r1 9724

This theorem is referenced by:  onvf1odlem4  35453  onvf1od  35454