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Theorem bnj1398 31401
 Description: Technical lemma for bnj60 31429. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1398.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1398.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1398.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1398.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1398.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1398.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1398.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1398.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1398.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1398.10 𝑃 = 𝐻
bnj1398.11 (𝜃 ↔ (𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
bnj1398.12 (𝜂 ↔ (𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
Assertion
Ref Expression
bnj1398 (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝐵,𝑓   𝑦,𝐶   𝑦,𝐷   𝑧,𝐻   𝑧,𝑃   𝑅,𝑓,𝑥,𝑦,𝑧   𝜒,𝑧   𝑓,𝑑,𝑥   𝜓,𝑦   𝜏,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑑)   𝐵(𝑥,𝑦,𝑧,𝑑)   𝐶(𝑥,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑑)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1398
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 bnj1398.11 . . . . 5 (𝜃 ↔ (𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
2 df-iun 4666 . . . . . . . . . 10 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = {𝑧 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))}
32bnj1436 31209 . . . . . . . . 9 (𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
41, 3simplbiim 661 . . . . . . . 8 (𝜃 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
5 bnj1398.12 . . . . . . . 8 (𝜂 ↔ (𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
6 bnj1398.7 . . . . . . . . . . . 12 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
7 nfv 1984 . . . . . . . . . . . . 13 𝑦𝜓
8 nfv 1984 . . . . . . . . . . . . 13 𝑦 𝑥𝐷
9 nfra1 3071 . . . . . . . . . . . . 13 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
107, 8, 9nf3an 1972 . . . . . . . . . . . 12 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
116, 10nfxfr 1920 . . . . . . . . . . 11 𝑦𝜒
12 nfiu1 4694 . . . . . . . . . . . 12 𝑦 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
1312nfcri 2888 . . . . . . . . . . 11 𝑦 𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
1411, 13nfan 1969 . . . . . . . . . 10 𝑦(𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
151, 14nfxfr 1920 . . . . . . . . 9 𝑦𝜃
1615nf5ri 2204 . . . . . . . 8 (𝜃 → ∀𝑦𝜃)
174, 5, 16bnj1521 31220 . . . . . . 7 (𝜃 → ∃𝑦𝜂)
18 bnj1398.6 . . . . . . . . . . . . . . . . . . 19 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
19 nfv 1984 . . . . . . . . . . . . . . . . . . . 20 𝑓 𝑅 FrSe 𝐴
20 bnj1398.5 . . . . . . . . . . . . . . . . . . . . . 22 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
21 nfe1 2168 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓𝑓𝜏
2221nfn 1925 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ¬ ∃𝑓𝜏
23 nfcv 2894 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓𝐴
2422, 23nfrab 3254 . . . . . . . . . . . . . . . . . . . . . 22 𝑓{𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2520, 24nfcxfr 2892 . . . . . . . . . . . . . . . . . . . . 21 𝑓𝐷
26 nfcv 2894 . . . . . . . . . . . . . . . . . . . . 21 𝑓
2725, 26nfne 3024 . . . . . . . . . . . . . . . . . . . 20 𝑓 𝐷 ≠ ∅
2819, 27nfan 1969 . . . . . . . . . . . . . . . . . . 19 𝑓(𝑅 FrSe 𝐴𝐷 ≠ ∅)
2918, 28nfxfr 1920 . . . . . . . . . . . . . . . . . 18 𝑓𝜓
3025nfcri 2888 . . . . . . . . . . . . . . . . . 18 𝑓 𝑥𝐷
31 nfv 1984 . . . . . . . . . . . . . . . . . . 19 𝑓 ¬ 𝑦𝑅𝑥
3225, 31nfral 3075 . . . . . . . . . . . . . . . . . 18 𝑓𝑦𝐷 ¬ 𝑦𝑅𝑥
3329, 30, 32nf3an 1972 . . . . . . . . . . . . . . . . 17 𝑓(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
346, 33nfxfr 1920 . . . . . . . . . . . . . . . 16 𝑓𝜒
35 nfv 1984 . . . . . . . . . . . . . . . 16 𝑓 𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
3634, 35nfan 1969 . . . . . . . . . . . . . . 15 𝑓(𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
371, 36nfxfr 1920 . . . . . . . . . . . . . 14 𝑓𝜃
38 nfv 1984 . . . . . . . . . . . . . 14 𝑓 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)
39 nfv 1984 . . . . . . . . . . . . . 14 𝑓 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
4037, 38, 39nf3an 1972 . . . . . . . . . . . . 13 𝑓(𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
415, 40nfxfr 1920 . . . . . . . . . . . 12 𝑓𝜂
4241nf5ri 2204 . . . . . . . . . . 11 (𝜂 → ∀𝑓𝜂)
431simplbi 478 . . . . . . . . . . . . 13 (𝜃𝜒)
445, 43bnj835 31128 . . . . . . . . . . . 12 (𝜂𝜒)
455simp2bi 1140 . . . . . . . . . . . 12 (𝜂𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
46 bnj1398.1 . . . . . . . . . . . . . 14 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
47 bnj1398.2 . . . . . . . . . . . . . 14 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
48 bnj1398.3 . . . . . . . . . . . . . 14 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
49 bnj1398.4 . . . . . . . . . . . . . 14 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
50 bnj1398.8 . . . . . . . . . . . . . 14 (𝜏′[𝑦 / 𝑥]𝜏)
5146, 47, 48, 49, 20, 18, 6, 50bnj1388 31400 . . . . . . . . . . . . 13 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
52 rsp 3059 . . . . . . . . . . . . 13 (∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′ → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
5351, 52syl 17 . . . . . . . . . . . 12 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
5444, 45, 53sylc 65 . . . . . . . . . . 11 (𝜂 → ∃𝑓𝜏′)
5542, 54bnj596 31115 . . . . . . . . . 10 (𝜂 → ∃𝑓(𝜂𝜏′))
5646, 47, 48, 49, 50bnj1373 31397 . . . . . . . . . . . . . 14 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
5756simplbi 478 . . . . . . . . . . . . 13 (𝜏′𝑓𝐶)
5857adantl 473 . . . . . . . . . . . 12 ((𝜂𝜏′) → 𝑓𝐶)
5956simprbi 483 . . . . . . . . . . . . 13 (𝜏′ → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
60 rspe 3133 . . . . . . . . . . . . 13 ((𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
6145, 59, 60syl2an 495 . . . . . . . . . . . 12 ((𝜂𝜏′) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
62 bnj1398.9 . . . . . . . . . . . . . 14 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
6362abeq2i 2865 . . . . . . . . . . . . 13 (𝑓𝐻 ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
6456rexbii 3171 . . . . . . . . . . . . 13 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
65 r19.42v 3222 . . . . . . . . . . . . 13 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
6663, 64, 653bitri 286 . . . . . . . . . . . 12 (𝑓𝐻 ↔ (𝑓𝐶 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
6758, 61, 66sylanbrc 701 . . . . . . . . . . 11 ((𝜂𝜏′) → 𝑓𝐻)
685simp3bi 1141 . . . . . . . . . . . . 13 (𝜂𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
6968adantr 472 . . . . . . . . . . . 12 ((𝜂𝜏′) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
7059adantl 473 . . . . . . . . . . . 12 ((𝜂𝜏′) → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
7169, 70eleqtrrd 2834 . . . . . . . . . . 11 ((𝜂𝜏′) → 𝑧 ∈ dom 𝑓)
7267, 71jca 555 . . . . . . . . . 10 ((𝜂𝜏′) → (𝑓𝐻𝑧 ∈ dom 𝑓))
7355, 72bnj593 31114 . . . . . . . . 9 (𝜂 → ∃𝑓(𝑓𝐻𝑧 ∈ dom 𝑓))
74 df-rex 3048 . . . . . . . . 9 (∃𝑓𝐻 𝑧 ∈ dom 𝑓 ↔ ∃𝑓(𝑓𝐻𝑧 ∈ dom 𝑓))
7573, 74sylibr 224 . . . . . . . 8 (𝜂 → ∃𝑓𝐻 𝑧 ∈ dom 𝑓)
76 bnj1398.10 . . . . . . . . . . . 12 𝑃 = 𝐻
7776dmeqi 5472 . . . . . . . . . . 11 dom 𝑃 = dom 𝐻
7862bnj1317 31191 . . . . . . . . . . . 12 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
7978bnj1400 31205 . . . . . . . . . . 11 dom 𝐻 = 𝑓𝐻 dom 𝑓
8077, 79eqtri 2774 . . . . . . . . . 10 dom 𝑃 = 𝑓𝐻 dom 𝑓
8180eleq2i 2823 . . . . . . . . 9 (𝑧 ∈ dom 𝑃𝑧 𝑓𝐻 dom 𝑓)
82 eliun 4668 . . . . . . . . 9 (𝑧 𝑓𝐻 dom 𝑓 ↔ ∃𝑓𝐻 𝑧 ∈ dom 𝑓)
8381, 82bitri 264 . . . . . . . 8 (𝑧 ∈ dom 𝑃 ↔ ∃𝑓𝐻 𝑧 ∈ dom 𝑓)
8475, 83sylibr 224 . . . . . . 7 (𝜂𝑧 ∈ dom 𝑃)
8517, 84bnj593 31114 . . . . . 6 (𝜃 → ∃𝑦 𝑧 ∈ dom 𝑃)
86 nfre1 3135 . . . . . . . . . . . 12 𝑦𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
8786nfab 2899 . . . . . . . . . . 11 𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
8862, 87nfcxfr 2892 . . . . . . . . . 10 𝑦𝐻
8988nfuni 4586 . . . . . . . . 9 𝑦 𝐻
9076, 89nfcxfr 2892 . . . . . . . 8 𝑦𝑃
9190nfdm 5514 . . . . . . 7 𝑦dom 𝑃
9291nfcrii 2887 . . . . . 6 (𝑧 ∈ dom 𝑃 → ∀𝑦 𝑧 ∈ dom 𝑃)
9385, 92bnj1397 31204 . . . . 5 (𝜃𝑧 ∈ dom 𝑃)
941, 93sylbir 225 . . . 4 ((𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ dom 𝑃)
9594ex 449 . . 3 (𝜒 → (𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ dom 𝑃))
9695ssrdv 3742 . 2 (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ⊆ dom 𝑃)
97 simpr 479 . . . . . . 7 ((𝑓𝐻𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
9866simprbi 483 . . . . . . . 8 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
9998adantr 472 . . . . . . 7 ((𝑓𝐻𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
100 r19.42v 3222 . . . . . . . 8 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝑧 ∈ dom 𝑓 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
101 eleq2 2820 . . . . . . . . . 10 (dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → (𝑧 ∈ dom 𝑓𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
102101biimpac 504 . . . . . . . . 9 ((𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
103102reximi 3141 . . . . . . . 8 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
104100, 103sylbir 225 . . . . . . 7 ((𝑧 ∈ dom 𝑓 ∧ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
10597, 99, 104syl2anc 696 . . . . . 6 ((𝑓𝐻𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
106105rexlimiva 3158 . . . . 5 (∃𝑓𝐻 𝑧 ∈ dom 𝑓 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
107 eliun 4668 . . . . 5 (𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
108106, 83, 1073imtr4i 281 . . . 4 (𝑧 ∈ dom 𝑃𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
109108ssriv 3740 . . 3 dom 𝑃 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
110109a1i 11 . 2 (𝜒 → dom 𝑃 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
11196, 110eqssd 3753 1 (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1624  ∃wex 1845   ∈ wcel 2131  {cab 2738   ≠ wne 2924  ∀wral 3042  ∃wrex 3043  {crab 3046  [wsbc 3568   ∪ cun 3705   ⊆ wss 3707  ∅c0 4050  {csn 4313  ⟨cop 4319  ∪ cuni 4580  ∪ ciun 4664   class class class wbr 4796  dom cdm 5258   ↾ cres 5260   Fn wfn 6036  ‘cfv 6041   predc-bnj14 31055   FrSe w-bnj15 31059   trClc-bnj18 31061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-dm 5268  df-bnj14 31056  df-bnj18 31062 This theorem is referenced by:  bnj1415  31405
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