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Theorem r0weon 9426
Description: A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
leweon.1 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
r0weon.1 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
Assertion
Ref Expression
r0weon (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Distinct variable groups:   𝑧,𝑤,𝐿   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧,𝑤)   𝐿(𝑥,𝑦)

Proof of Theorem r0weon
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 r0weon.1 . . . . 5 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
2 fveq2 6663 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
3 fveq2 6663 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (2nd𝑥) = (2nd𝑧))
42, 3uneq12d 4137 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑧) ∪ (2nd𝑧)))
5 eqid 2818 . . . . . . . . . . 11 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
6 fvex 6676 . . . . . . . . . . . 12 (1st𝑧) ∈ V
7 fvex 6676 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
86, 7unex 7458 . . . . . . . . . . 11 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
94, 5, 8fvmpt 6761 . . . . . . . . . 10 (𝑧 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((1st𝑧) ∪ (2nd𝑧)))
10 fveq2 6663 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (1st𝑥) = (1st𝑤))
11 fveq2 6663 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (2nd𝑥) = (2nd𝑤))
1210, 11uneq12d 4137 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑤) ∪ (2nd𝑤)))
13 fvex 6676 . . . . . . . . . . . 12 (1st𝑤) ∈ V
14 fvex 6676 . . . . . . . . . . . 12 (2nd𝑤) ∈ V
1513, 14unex 7458 . . . . . . . . . . 11 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
1612, 5, 15fvmpt 6761 . . . . . . . . . 10 (𝑤 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) = ((1st𝑤) ∪ (2nd𝑤)))
179, 16breqan12d 5073 . . . . . . . . 9 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) E ((1st𝑤) ∪ (2nd𝑤))))
1815epeli 5461 . . . . . . . . 9 (((1st𝑧) ∪ (2nd𝑧)) E ((1st𝑤) ∪ (2nd𝑤)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)))
1917, 18syl6bb 288 . . . . . . . 8 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤))))
209, 16eqeqan12d 2835 . . . . . . . . 9 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
2120anbi1d 629 . . . . . . . 8 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤) ↔ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
2219, 21orbi12d 912 . . . . . . 7 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → ((((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
2322pm5.32i 575 . . . . . 6 (((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤))) ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
2423opabbii 5124 . . . . 5 {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
251, 24eqtr4i 2844 . . . 4 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) E ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∨ (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ∧ 𝑧𝐿𝑤)))}
26 xp1st 7710 . . . . . . . 8 (𝑥 ∈ (On × On) → (1st𝑥) ∈ On)
27 xp2nd 7711 . . . . . . . 8 (𝑥 ∈ (On × On) → (2nd𝑥) ∈ On)
28 fvex 6676 . . . . . . . . . 10 (1st𝑥) ∈ V
2928elon 6193 . . . . . . . . 9 ((1st𝑥) ∈ On ↔ Ord (1st𝑥))
30 fvex 6676 . . . . . . . . . 10 (2nd𝑥) ∈ V
3130elon 6193 . . . . . . . . 9 ((2nd𝑥) ∈ On ↔ Ord (2nd𝑥))
32 ordun 6285 . . . . . . . . 9 ((Ord (1st𝑥) ∧ Ord (2nd𝑥)) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3329, 31, 32syl2anb 597 . . . . . . . 8 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3426, 27, 33syl2anc 584 . . . . . . 7 (𝑥 ∈ (On × On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3528, 30unex 7458 . . . . . . . 8 ((1st𝑥) ∪ (2nd𝑥)) ∈ V
3635elon 6193 . . . . . . 7 (((1st𝑥) ∪ (2nd𝑥)) ∈ On ↔ Ord ((1st𝑥) ∪ (2nd𝑥)))
3734, 36sylibr 235 . . . . . 6 (𝑥 ∈ (On × On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ On)
385, 37fmpti 6868 . . . . 5 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On
3938a1i 11 . . . 4 (⊤ → (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On)
40 epweon 7486 . . . . 5 E We On
4140a1i 11 . . . 4 (⊤ → E We On)
42 leweon.1 . . . . . 6 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
4342leweon 9425 . . . . 5 𝐿 We (On × On)
4443a1i 11 . . . 4 (⊤ → 𝐿 We (On × On))
45 vex 3495 . . . . . . . 8 𝑢 ∈ V
4645dmex 7605 . . . . . . 7 dom 𝑢 ∈ V
4745rnex 7606 . . . . . . 7 ran 𝑢 ∈ V
4846, 47unex 7458 . . . . . 6 (dom 𝑢 ∪ ran 𝑢) ∈ V
49 imadmres 6084 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢)
50 inss2 4203 . . . . . . . . . 10 (𝑢 ∩ (On × On)) ⊆ (On × On)
51 ssun1 4145 . . . . . . . . . . . . . 14 dom 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
52 elinel2 4170 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ (On × On))
53 1st2nd2 7717 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (On × On) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
5452, 53syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
55 elinel1 4169 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥𝑢)
5654, 55eqeltrrd 2911 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑢 ∩ (On × On)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢)
5728, 30opeldm 5769 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (1st𝑥) ∈ dom 𝑢)
5856, 57syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ dom 𝑢)
5951, 58sseldi 3962 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
60 ssun2 4146 . . . . . . . . . . . . . 14 ran 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
6128, 30opelrn 5806 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (2nd𝑥) ∈ ran 𝑢)
6256, 61syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ ran 𝑢)
6360, 62sseldi 3962 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
6459, 63prssd 4747 . . . . . . . . . . . 12 (𝑥 ∈ (𝑢 ∩ (On × On)) → {(1st𝑥), (2nd𝑥)} ⊆ (dom 𝑢 ∪ ran 𝑢))
6552, 26syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ On)
6652, 27syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ On)
67 ordunpr 7530 . . . . . . . . . . . . 13 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6865, 66, 67syl2anc 584 . . . . . . . . . . . 12 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6964, 68sseldd 3965 . . . . . . . . . . 11 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢))
7069rgen 3145 . . . . . . . . . 10 𝑥 ∈ (𝑢 ∩ (On × On))((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)
71 ssrab 4046 . . . . . . . . . 10 ((𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)} ↔ ((𝑢 ∩ (On × On)) ⊆ (On × On) ∧ ∀𝑥 ∈ (𝑢 ∩ (On × On))((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)))
7250, 70, 71mpbir2an 707 . . . . . . . . 9 (𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
73 dmres 5868 . . . . . . . . . 10 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))))
7438fdmi 6517 . . . . . . . . . . 11 dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (On × On)
7574ineq2i 4183 . . . . . . . . . 10 (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))) = (𝑢 ∩ (On × On))
7673, 75eqtri 2841 . . . . . . . . 9 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ (On × On))
775mptpreima 6085 . . . . . . . . 9 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢)) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
7872, 76, 773sstr4i 4007 . . . . . . . 8 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢))
79 funmpt 6386 . . . . . . . . 9 Fun (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
80 resss 5871 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
81 dmss 5764 . . . . . . . . . 10 (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) → dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))))
8280, 81ax-mp 5 . . . . . . . . 9 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
83 funimass3 6816 . . . . . . . . 9 ((Fun (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ∧ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢))))
8479, 82, 83mp2an 688 . . . . . . . 8 (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢) ↔ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢)))
8578, 84mpbir 232 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢)
8649, 85eqsstrri 3999 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (dom 𝑢 ∪ ran 𝑢)
8748, 86ssexi 5217 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
8887a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
8925, 39, 41, 44, 88fnwe 7815 . . 3 (⊤ → 𝑅 We (On × On))
90 epse 5531 . . . . 5 E Se On
9190a1i 11 . . . 4 (⊤ → E Se On)
92 vuniex 7455 . . . . . . . 8 𝑢 ∈ V
9392pwex 5272 . . . . . . 7 𝒫 𝑢 ∈ V
9493, 93xpex 7465 . . . . . 6 (𝒫 𝑢 × 𝒫 𝑢) ∈ V
955mptpreima 6085 . . . . . . . 8 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢}
96 df-rab 3144 . . . . . . . 8 {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢} = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9795, 96eqtri 2841 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9853adantr 481 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
99 elssuni 4859 . . . . . . . . . . . . 13 (((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢 → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
10099adantl 482 . . . . . . . . . . . 12 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
101100unssad 4160 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ⊆ 𝑢)
10228elpw 4542 . . . . . . . . . . 11 ((1st𝑥) ∈ 𝒫 𝑢 ↔ (1st𝑥) ⊆ 𝑢)
103101, 102sylibr 235 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ∈ 𝒫 𝑢)
104100unssbd 4161 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ⊆ 𝑢)
10530elpw 4542 . . . . . . . . . . 11 ((2nd𝑥) ∈ 𝒫 𝑢 ↔ (2nd𝑥) ⊆ 𝑢)
106104, 105sylibr 235 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ∈ 𝒫 𝑢)
107103, 106jca 512 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢))
108 elxp6 7712 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢)))
10998, 107, 108sylanbrc 583 . . . . . . . 8 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢))
110109abssi 4043 . . . . . . 7 {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)} ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11197, 110eqsstri 3998 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11294, 111ssexi 5217 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
113112a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
11425, 39, 91, 113fnse 7816 . . 3 (⊤ → 𝑅 Se (On × On))
11589, 114jca 512 . 2 (⊤ → (𝑅 We (On × On) ∧ 𝑅 Se (On × On)))
116115mptru 1535 1 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841   = wceq 1528  wtru 1529  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492  cun 3931  cin 3932  wss 3933  𝒫 cpw 4535  {cpr 4559  cop 4563   cuni 4830   class class class wbr 5057  {copab 5119  cmpt 5137   E cep 5457   Se wse 5505   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Ord word 6183  Oncon0 6184  Fun wfun 6342  wf 6344  cfv 6348  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-1st 7678  df-2nd 7679
This theorem is referenced by:  infxpenlem  9427
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