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Theorem wfisgOLD 6356
Description: Obsolete proof of wfisg 6355 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 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
wfisgOLD ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 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.
StepHypRef Expression
1 ssrab2 4078 . . 3 {𝑦𝐴𝜑} ⊆ 𝐴
2 dfss3 3971 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑})
3 nfcv 2904 . . . . . . . . . . 11 𝑦𝐴
43elrabsf 3826 . . . . . . . . . 10 (𝑧 ∈ {𝑦𝐴𝜑} ↔ (𝑧𝐴[𝑧 / 𝑦]𝜑))
54simprbi 498 . . . . . . . . 9 (𝑧 ∈ {𝑦𝐴𝜑} → [𝑧 / 𝑦]𝜑)
65ralimi 3084 . . . . . . . 8 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
72, 6sylbi 216 . . . . . . 7 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8 nfv 1918 . . . . . . . . 9 𝑦 𝑤𝐴
9 nfcv 2904 . . . . . . . . . . 11 𝑦Pred(𝑅, 𝐴, 𝑤)
10 nfsbc1v 3798 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
119, 10nfralw 3309 . . . . . . . . . 10 𝑦𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12 nfsbc1v 3798 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
1311, 12nfim 1900 . . . . . . . . 9 𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)
148, 13nfim 1900 . . . . . . . 8 𝑦(𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
15 eleq1w 2817 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
16 predeq3 6305 . . . . . . . . . . 11 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1716raleqdv 3326 . . . . . . . . . 10 (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18 sbceq1a 3789 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝜑[𝑤 / 𝑦]𝜑))
1917, 18imbi12d 345 . . . . . . . . 9 (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)))
2015, 19imbi12d 345 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑)) ↔ (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))))
21 wfisgOLD.1 . . . . . . . 8 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
2214, 20, 21chvarfv 2234 . . . . . . 7 (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
237, 22syl5 34 . . . . . 6 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → [𝑤 / 𝑦]𝜑))
2423anc2li 557 . . . . 5 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → (𝑤𝐴[𝑤 / 𝑦]𝜑)))
253elrabsf 3826 . . . . 5 (𝑤 ∈ {𝑦𝐴𝜑} ↔ (𝑤𝐴[𝑤 / 𝑦]𝜑))
2624, 25imbitrrdi 251 . . . 4 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
2726rgen 3064 . . 3 𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑})
28 wfi 6352 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))) → 𝐴 = {𝑦𝐴𝜑})
291, 27, 28mpanr12 704 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐴 = {𝑦𝐴𝜑})
30 rabid2 3465 . 2 (𝐴 = {𝑦𝐴𝜑} ↔ ∀𝑦𝐴 𝜑)
3129, 30sylib 217 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  {crab 3433  [wsbc 3778  wss 3949   Se wse 5630   We wwe 5631  Predcpred 6300
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301
This theorem is referenced by: (None)
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