Theorem wfisgOLD 6375

Description: Obsolete version of wfisg 6374 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.)

Hypothesis

Ref	Expression
wfisgOLD.1	⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))

Assertion

Ref	Expression
wfisgOLD	⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wfisgOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4080	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		dfss3 3972	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
3		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
4	3	elrabsf 3834	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜑))
5	4	simprbi 496	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑧 / 𝑦]𝜑)
6	5	ralimi 3083	. . . . . . . 8 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7	2, 6	sylbi 217	. . . . . . 7 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
9		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑤)
10		nfsbc1v 3808	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
11	9, 10	nfralw 3311	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12		nfsbc1v 3808	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
13	11, 12	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)
14	8, 13	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
15		eleq1w 2824	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
16		predeq3 6325	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
17	16	raleqdv 3326	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18		sbceq1a 3799	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
19	17, 18	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)))
20	15, 19	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ↔ (𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))))
21		wfisgOLD.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))
22	14, 20, 21	chvarfv 2240	. . . . . . 7 ⊢ (𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
23	7, 22	syl5 34	. . . . . 6 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑤 / 𝑦]𝜑))
24	23	anc2li 555	. . . . 5 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑)))
25	3	elrabsf 3834	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑))
26	24, 25	imbitrrdi 252	. . . 4 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))
27	26	rgen 3063	. . 3 ⊢ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
28		wfi 6371	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ ({𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
29	1, 27, 28	mpanr12 705	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
30		rabid2 3470	. 2 ⊢ (𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝐴 𝜑)
31	29, 30	sylib 218	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  [wsbc 3788   ⊆ wss 3951   Se wse 5635   We wwe 5636  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321

This theorem is referenced by: (None)