Theorem onvf1odlem2 35300

Description: Lemma for onvf1od 35303. (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 35299	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴)
3		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑅1
4		nfrab1 3420	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
5	4	nfint 4913	. . . . . . . . 9 ⊢ Ⅎ𝑥∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}
6	3, 5	nffv 6845	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
7		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ 𝑣 ∈ 𝐴
8	6, 7	nfrexw 3285	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑣 ∈ (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}) ¬ 𝑣 ∈ 𝐴
9		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ 𝐴 ↔ 𝑣 ∈ 𝐴))
10	9	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (¬ 𝑦 ∈ 𝐴 ↔ ¬ 𝑣 ∈ 𝐴))
11	10	cbvrexvw 3216	. . . . . . . 8 ⊢ (∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴 ↔ ∃𝑣 ∈ (𝑅1‘𝑥) ¬ 𝑣 ∈ 𝐴)
12		fveq2 6835	. . . . . . . . 9 ⊢ (𝑥 = ∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴} → (𝑅1‘𝑥) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴}))
13	12	rexeqdv 3298	. . . . . . . 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 7741	. . . . . 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 6838	. . . . . . 7 ⊢ (𝑅1‘𝑀) = (𝑅1‘∩ {𝑥 ∈ On ∣ ∃𝑦 ∈ (𝑅1‘𝑥) ¬ 𝑦 ∈ 𝐴})
18	17	rexeqi 3296	. . . . . 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 3062	. . . . 5 ⊢ (∃𝑣 ∈ (𝑅1‘𝑀) ¬ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
22		nss 3999	. . . . 5 ⊢ (¬ (𝑅1‘𝑀) ⊆ 𝐴 ↔ ∃𝑣(𝑣 ∈ (𝑅1‘𝑀) ∧ ¬ 𝑣 ∈ 𝐴))
23		ssdif0 4319	. . . . . 6 ⊢ ((𝑅1‘𝑀) ⊆ 𝐴 ↔ ((𝑅1‘𝑀) ∖ 𝐴) = ∅)
24	23	necon3bbii 2980	. . . . 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 6848	. . . . 5 ⊢ (𝑅1‘𝑀) ∈ V
28	27	difexi 5276	. . . 4 ⊢ ((𝑅1‘𝑀) ∖ 𝐴) ∈ V
29		neeq1 2995	. . . . 5 ⊢ (𝑧 = ((𝑅1‘𝑀) ∖ 𝐴) → (𝑧 ≠ ∅ ↔ ((𝑅1‘𝑀) ∖ 𝐴) ≠ ∅))
30		fveq2 6835	. . . . . 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 2831	. . . . 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 3560	. . 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 2828	. 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3061  {crab 3400   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4286  ∩ cint 4903  Oncon0 6318  ‘cfv 6493  𝑅₁cr1 9678

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680

This theorem is referenced by:  onvf1odlem4  35302  onvf1od  35303