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Theorem bnj1398 30837
Description: Technical lemma for bnj60 30865. 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 4492 . . . . . . . . . 10 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = {𝑧 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))}
32bnj1436 30645 . . . . . . . . 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 658 . . . . . . . 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 1840 . . . . . . . . . . . . 13 𝑦𝜓
8 nfv 1840 . . . . . . . . . . . . 13 𝑦 𝑥𝐷
9 nfra1 2936 . . . . . . . . . . . . 13 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
107, 8, 9nf3an 1828 . . . . . . . . . . . 12 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
116, 10nfxfr 1776 . . . . . . . . . . 11 𝑦𝜒
12 nfiu1 4521 . . . . . . . . . . . 12 𝑦 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
1312nfcri 2755 . . . . . . . . . . 11 𝑦 𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
1411, 13nfan 1825 . . . . . . . . . 10 𝑦(𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
151, 14nfxfr 1776 . . . . . . . . 9 𝑦𝜃
1615nf5ri 2063 . . . . . . . 8 (𝜃 → ∀𝑦𝜃)
174, 5, 16bnj1521 30656 . . . . . . 7 (𝜃 → ∃𝑦𝜂)
18 bnj1398.6 . . . . . . . . . . . . . . . . . . 19 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
19 nfv 1840 . . . . . . . . . . . . . . . . . . . 20 𝑓 𝑅 FrSe 𝐴
20 bnj1398.5 . . . . . . . . . . . . . . . . . . . . . 22 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
21 nfe1 2024 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑓𝑓𝜏
2221nfn 1781 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓 ¬ ∃𝑓𝜏
23 nfcv 2761 . . . . . . . . . . . . . . . . . . . . . . 23 𝑓𝐴
2422, 23nfrab 3115 . . . . . . . . . . . . . . . . . . . . . 22 𝑓{𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2520, 24nfcxfr 2759 . . . . . . . . . . . . . . . . . . . . 21 𝑓𝐷
26 nfcv 2761 . . . . . . . . . . . . . . . . . . . . 21 𝑓
2725, 26nfne 2890 . . . . . . . . . . . . . . . . . . . 20 𝑓 𝐷 ≠ ∅
2819, 27nfan 1825 . . . . . . . . . . . . . . . . . . 19 𝑓(𝑅 FrSe 𝐴𝐷 ≠ ∅)
2918, 28nfxfr 1776 . . . . . . . . . . . . . . . . . 18 𝑓𝜓
3025nfcri 2755 . . . . . . . . . . . . . . . . . 18 𝑓 𝑥𝐷
31 nfv 1840 . . . . . . . . . . . . . . . . . . 19 𝑓 ¬ 𝑦𝑅𝑥
3225, 31nfral 2940 . . . . . . . . . . . . . . . . . 18 𝑓𝑦𝐷 ¬ 𝑦𝑅𝑥
3329, 30, 32nf3an 1828 . . . . . . . . . . . . . . . . 17 𝑓(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
346, 33nfxfr 1776 . . . . . . . . . . . . . . . 16 𝑓𝜒
35 nfv 1840 . . . . . . . . . . . . . . . 16 𝑓 𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
3634, 35nfan 1825 . . . . . . . . . . . . . . 15 𝑓(𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
371, 36nfxfr 1776 . . . . . . . . . . . . . 14 𝑓𝜃
38 nfv 1840 . . . . . . . . . . . . . 14 𝑓 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)
39 nfv 1840 . . . . . . . . . . . . . 14 𝑓 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))
4037, 38, 39nf3an 1828 . . . . . . . . . . . . 13 𝑓(𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
415, 40nfxfr 1776 . . . . . . . . . . . 12 𝑓𝜂
4241nf5ri 2063 . . . . . . . . . . 11 (𝜂 → ∀𝑓𝜂)
431simplbi 476 . . . . . . . . . . . . 13 (𝜃𝜒)
445, 43bnj835 30564 . . . . . . . . . . . 12 (𝜂𝜒)
455simp2bi 1075 . . . . . . . . . . . 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 30836 . . . . . . . . . . . . 13 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
52 rsp 2924 . . . . . . . . . . . . 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 30551 . . . . . . . . . 10 (𝜂 → ∃𝑓(𝜂𝜏′))
5646, 47, 48, 49, 50bnj1373 30833 . . . . . . . . . . . . . 14 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
5756simplbi 476 . . . . . . . . . . . . 13 (𝜏′𝑓𝐶)
5857adantl 482 . . . . . . . . . . . 12 ((𝜂𝜏′) → 𝑓𝐶)
5956simprbi 480 . . . . . . . . . . . . 13 (𝜏′ → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
60 rspe 2998 . . . . . . . . . . . . 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 494 . . . . . . . . . . . 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 2732 . . . . . . . . . . . . 13 (𝑓𝐻 ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
6456rexbii 3035 . . . . . . . . . . . . 13 (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)(𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
65 r19.42v 3085 . . . . . . . . . . . . 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 697 . . . . . . . . . . 11 ((𝜂𝜏′) → 𝑓𝐻)
685simp3bi 1076 . . . . . . . . . . . . 13 (𝜂𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
6968adantr 481 . . . . . . . . . . . 12 ((𝜂𝜏′) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
7059adantl 482 . . . . . . . . . . . 12 ((𝜂𝜏′) → dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
7169, 70eleqtrrd 2701 . . . . . . . . . . 11 ((𝜂𝜏′) → 𝑧 ∈ dom 𝑓)
7267, 71jca 554 . . . . . . . . . 10 ((𝜂𝜏′) → (𝑓𝐻𝑧 ∈ dom 𝑓))
7355, 72bnj593 30550 . . . . . . . . 9 (𝜂 → ∃𝑓(𝑓𝐻𝑧 ∈ dom 𝑓))
74 df-rex 2913 . . . . . . . . 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 5290 . . . . . . . . . . 11 dom 𝑃 = dom 𝐻
7862bnj1317 30627 . . . . . . . . . . . 12 (𝑤𝐻 → ∀𝑓 𝑤𝐻)
7978bnj1400 30641 . . . . . . . . . . 11 dom 𝐻 = 𝑓𝐻 dom 𝑓
8077, 79eqtri 2643 . . . . . . . . . 10 dom 𝑃 = 𝑓𝐻 dom 𝑓
8180eleq2i 2690 . . . . . . . . 9 (𝑧 ∈ dom 𝑃𝑧 𝑓𝐻 dom 𝑓)
82 eliun 4495 . . . . . . . . 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 30550 . . . . . 6 (𝜃 → ∃𝑦 𝑧 ∈ dom 𝑃)
86 nfre1 3000 . . . . . . . . . . . 12 𝑦𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
8786nfab 2765 . . . . . . . . . . 11 𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
8862, 87nfcxfr 2759 . . . . . . . . . 10 𝑦𝐻
8988nfuni 4413 . . . . . . . . 9 𝑦 𝐻
9076, 89nfcxfr 2759 . . . . . . . 8 𝑦𝑃
9190nfdm 5332 . . . . . . 7 𝑦dom 𝑃
9291nfcrii 2754 . . . . . 6 (𝑧 ∈ dom 𝑃 → ∀𝑦 𝑧 ∈ dom 𝑃)
9385, 92bnj1397 30640 . . . . 5 (𝜃𝑧 ∈ dom 𝑃)
941, 93sylbir 225 . . . 4 ((𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ dom 𝑃)
9594ex 450 . . 3 (𝜒 → (𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ dom 𝑃))
9695ssrdv 3593 . 2 (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) ⊆ dom 𝑃)
97 simpr 477 . . . . . . 7 ((𝑓𝐻𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
9866simprbi 480 . . . . . . . 8 (𝑓𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
9998adantr 481 . . . . . . 7 ((𝑓𝐻𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
100 r19.42v 3085 . . . . . . . 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 2687 . . . . . . . . . 10 (dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → (𝑧 ∈ dom 𝑓𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
102101biimpac 503 . . . . . . . . 9 ((𝑧 ∈ dom 𝑓 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
103102reximi 3006 . . . . . . . 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 692 . . . . . 6 ((𝑓𝐻𝑧 ∈ dom 𝑓) → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
106105rexlimiva 3022 . . . . 5 (∃𝑓𝐻 𝑧 ∈ dom 𝑓 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
107 eliun 4495 . . . . 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 3591 . . 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 3604 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  [wsbc 3421  cun 3557  wss 3559  c0 3896  {csn 4153  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618  dom cdm 5079  cres 5081   Fn wfn 5847  cfv 5852   predc-bnj14 30488   FrSe w-bnj15 30492   trClc-bnj18 30494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-dm 5089  df-bnj14 30489  df-bnj18 30495
This theorem is referenced by:  bnj1415  30841
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