Theorem wfisgOLD 6257

Description: Obsolete proof of wfisg 6256 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 4013	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
2		dfss3 3909	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
3		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
4	3	elrabsf 3764	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜑))
5	4	simprbi 497	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → [𝑧 / 𝑦]𝜑)
6	5	ralimi 3087	. . . . . . . 8 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
7	2, 6	sylbi 216	. . . . . . 7 ⊢ (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
9		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑤)
10		nfsbc1v 3736	. . . . . . . . . . 11 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
11	9, 10	nfralw 3151	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12		nfsbc1v 3736	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
13	11, 12	nfim 1899	. . . . . . . . 9 ⊢ Ⅎ𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑)
14	8, 13	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑 → [𝑤 / 𝑦]𝜑))
15		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
16		predeq3 6206	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
17	16	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18		sbceq1a 3727	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
19	17, 18	imbi12d 345	. . . . . . . . 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 345	. . . . . . . 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 2233	. . . . . . 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 556	. . . . 5 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑)))
25	3	elrabsf 3764	. . . . 5 ⊢ (𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑦]𝜑))
26	24, 25	syl6ibr 251	. . . 4 ⊢ (𝑤 ∈ 𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}))
27	26	rgen 3074	. . 3 ⊢ ∀𝑤 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦 ∈ 𝐴 ∣ 𝜑} → 𝑤 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑})
28		wfi 6253	. . 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 702	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑})
30		rabid2 3314	. 2 ⊢ (𝐴 = {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑦 ∈ 𝐴 𝜑)
31	29, 30	sylib 217	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  [wsbc 3716   ⊆ wss 3887   Se wse 5542   We wwe 5543  Predcpred 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202

This theorem is referenced by: (None)