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Theorem r0weon 9427
 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 6649 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (1st𝑥) = (1st𝑧))
3 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (2nd𝑥) = (2nd𝑧))
42, 3uneq12d 4094 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑧) ∪ (2nd𝑧)))
5 eqid 2801 . . . . . . . . . . 11 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
6 fvex 6662 . . . . . . . . . . . 12 (1st𝑧) ∈ V
7 fvex 6662 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
86, 7unex 7453 . . . . . . . . . . 11 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
94, 5, 8fvmpt 6749 . . . . . . . . . 10 (𝑧 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((1st𝑧) ∪ (2nd𝑧)))
10 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (1st𝑥) = (1st𝑤))
11 fveq2 6649 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (2nd𝑥) = (2nd𝑤))
1210, 11uneq12d 4094 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((1st𝑥) ∪ (2nd𝑥)) = ((1st𝑤) ∪ (2nd𝑤)))
13 fvex 6662 . . . . . . . . . . . 12 (1st𝑤) ∈ V
14 fvex 6662 . . . . . . . . . . . 12 (2nd𝑤) ∈ V
1513, 14unex 7453 . . . . . . . . . . 11 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
1612, 5, 15fvmpt 6749 . . . . . . . . . 10 (𝑤 ∈ (On × On) → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) = ((1st𝑤) ∪ (2nd𝑤)))
179, 16breqan12d 5049 . . . . . . . . 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 5436 . . . . . . . . 9 (((1st𝑧) ∪ (2nd𝑧)) E ((1st𝑤) ∪ (2nd𝑤)) ↔ ((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)))
1917, 18syl6bb 290 . . . . . . . 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 2818 . . . . . . . . 9 ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) → (((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑧) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))‘𝑤) ↔ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
2120anbi1d 632 . . . . . . . 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 916 . . . . . . 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 578 . . . . . 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 5100 . . . . 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 2827 . . . 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 7707 . . . . . . . 8 (𝑥 ∈ (On × On) → (1st𝑥) ∈ On)
27 xp2nd 7708 . . . . . . . 8 (𝑥 ∈ (On × On) → (2nd𝑥) ∈ On)
28 fvex 6662 . . . . . . . . . 10 (1st𝑥) ∈ V
2928elon 6172 . . . . . . . . 9 ((1st𝑥) ∈ On ↔ Ord (1st𝑥))
30 fvex 6662 . . . . . . . . . 10 (2nd𝑥) ∈ V
3130elon 6172 . . . . . . . . 9 ((2nd𝑥) ∈ On ↔ Ord (2nd𝑥))
32 ordun 6264 . . . . . . . . 9 ((Ord (1st𝑥) ∧ Ord (2nd𝑥)) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3329, 31, 32syl2anb 600 . . . . . . . 8 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3426, 27, 33syl2anc 587 . . . . . . 7 (𝑥 ∈ (On × On) → Ord ((1st𝑥) ∪ (2nd𝑥)))
3528, 30unex 7453 . . . . . . . 8 ((1st𝑥) ∪ (2nd𝑥)) ∈ V
3635elon 6172 . . . . . . 7 (((1st𝑥) ∪ (2nd𝑥)) ∈ On ↔ Ord ((1st𝑥) ∪ (2nd𝑥)))
3734, 36sylibr 237 . . . . . 6 (𝑥 ∈ (On × On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ On)
385, 37fmpti 6857 . . . . 5 (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On
3938a1i 11 . . . 4 (⊤ → (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))):(On × On)⟶On)
40 epweon 7481 . . . . 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 9426 . . . . 5 𝐿 We (On × On)
4443a1i 11 . . . 4 (⊤ → 𝐿 We (On × On))
45 vex 3447 . . . . . . . 8 𝑢 ∈ V
4645dmex 7602 . . . . . . 7 dom 𝑢 ∈ V
4745rnex 7603 . . . . . . 7 ran 𝑢 ∈ V
4846, 47unex 7453 . . . . . 6 (dom 𝑢 ∪ ran 𝑢) ∈ V
49 imadmres 6062 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) = ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢)
50 inss2 4159 . . . . . . . . . 10 (𝑢 ∩ (On × On)) ⊆ (On × On)
51 ssun1 4102 . . . . . . . . . . . . . 14 dom 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
52 elinel2 4126 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 ∈ (On × On))
53 1st2nd2 7714 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (On × On) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
5452, 53syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
55 elinel1 4125 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝑢 ∩ (On × On)) → 𝑥𝑢)
5654, 55eqeltrrd 2894 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑢 ∩ (On × On)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢)
5728, 30opeldm 5744 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (1st𝑥) ∈ dom 𝑢)
5856, 57syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ dom 𝑢)
5951, 58sseldi 3916 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (1st𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
60 ssun2 4103 . . . . . . . . . . . . . 14 ran 𝑢 ⊆ (dom 𝑢 ∪ ran 𝑢)
6128, 30opelrn 5781 . . . . . . . . . . . . . . 15 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝑢 → (2nd𝑥) ∈ ran 𝑢)
6256, 61syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ ran 𝑢)
6360, 62sseldi 3916 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 ∩ (On × On)) → (2nd𝑥) ∈ (dom 𝑢 ∪ ran 𝑢))
6459, 63prssd 4718 . . . . . . . . . . . 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 7525 . . . . . . . . . . . . 13 (((1st𝑥) ∈ On ∧ (2nd𝑥) ∈ On) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6865, 66, 67syl2anc 587 . . . . . . . . . . . 12 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ {(1st𝑥), (2nd𝑥)})
6964, 68sseldd 3919 . . . . . . . . . . 11 (𝑥 ∈ (𝑢 ∩ (On × On)) → ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢))
7069rgen 3119 . . . . . . . . . 10 𝑥 ∈ (𝑢 ∩ (On × On))((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)
71 ssrab 4003 . . . . . . . . . 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 710 . . . . . . . . 9 (𝑢 ∩ (On × On)) ⊆ {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
73 dmres 5844 . . . . . . . . . 10 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))))
7438fdmi 6502 . . . . . . . . . . 11 dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) = (On × On)
7574ineq2i 4139 . . . . . . . . . 10 (𝑢 ∩ dom (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))) = (𝑢 ∩ (On × On))
7673, 75eqtri 2824 . . . . . . . . 9 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) = (𝑢 ∩ (On × On))
775mptpreima 6063 . . . . . . . . 9 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢)) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ (dom 𝑢 ∪ ran 𝑢)}
7872, 76, 773sstr4i 3961 . . . . . . . 8 dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ (dom 𝑢 ∪ ran 𝑢))
79 funmpt 6366 . . . . . . . . 9 Fun (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
80 resss 5847 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢) ⊆ (𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥)))
81 dmss 5739 . . . . . . . . . 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 6805 . . . . . . . . 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 691 . . . . . . . 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 234 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ dom ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) ↾ 𝑢)) ⊆ (dom 𝑢 ∪ ran 𝑢)
8649, 85eqsstrri 3953 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (dom 𝑢 ∪ ran 𝑢)
8748, 86ssexi 5193 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
8887a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
8925, 39, 41, 44, 88fnwe 7813 . . 3 (⊤ → 𝑅 We (On × On))
90 epse 5506 . . . . 5 E Se On
9190a1i 11 . . . 4 (⊤ → E Se On)
92 vuniex 7449 . . . . . . . 8 𝑢 ∈ V
9392pwex 5249 . . . . . . 7 𝒫 𝑢 ∈ V
9493, 93xpex 7460 . . . . . 6 (𝒫 𝑢 × 𝒫 𝑢) ∈ V
955mptpreima 6063 . . . . . . . 8 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢}
96 df-rab 3118 . . . . . . . 8 {𝑥 ∈ (On × On) ∣ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢} = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9795, 96eqtri 2824 . . . . . . 7 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) = {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)}
9853adantr 484 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
99 elssuni 4833 . . . . . . . . . . . . 13 (((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢 → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
10099adantl 485 . . . . . . . . . . . 12 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑢)
101100unssad 4117 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ⊆ 𝑢)
10228elpw 4504 . . . . . . . . . . 11 ((1st𝑥) ∈ 𝒫 𝑢 ↔ (1st𝑥) ⊆ 𝑢)
103101, 102sylibr 237 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (1st𝑥) ∈ 𝒫 𝑢)
104100unssbd 4118 . . . . . . . . . . 11 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ⊆ 𝑢)
10530elpw 4504 . . . . . . . . . . 11 ((2nd𝑥) ∈ 𝒫 𝑢 ↔ (2nd𝑥) ⊆ 𝑢)
106104, 105sylibr 237 . . . . . . . . . 10 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → (2nd𝑥) ∈ 𝒫 𝑢)
107103, 106jca 515 . . . . . . . . 9 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢))
108 elxp6 7709 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ 𝒫 𝑢 ∧ (2nd𝑥) ∈ 𝒫 𝑢)))
10998, 107, 108sylanbrc 586 . . . . . . . 8 ((𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢) → 𝑥 ∈ (𝒫 𝑢 × 𝒫 𝑢))
110109abssi 4000 . . . . . . 7 {𝑥 ∣ (𝑥 ∈ (On × On) ∧ ((1st𝑥) ∪ (2nd𝑥)) ∈ 𝑢)} ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11197, 110eqsstri 3952 . . . . . 6 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ⊆ (𝒫 𝑢 × 𝒫 𝑢)
11294, 111ssexi 5193 . . . . 5 ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V
113112a1i 11 . . . 4 (⊤ → ((𝑥 ∈ (On × On) ↦ ((1st𝑥) ∪ (2nd𝑥))) “ 𝑢) ∈ V)
11425, 39, 91, 113fnse 7814 . . 3 (⊤ → 𝑅 Se (On × On))
11589, 114jca 515 . 2 (⊤ → (𝑅 We (On × On) ∧ 𝑅 Se (On × On)))
116115mptru 1545 1 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ⊤wtru 1539   ∈ wcel 2112  {cab 2779  ∀wral 3109  {crab 3113  Vcvv 3444   ∪ cun 3882   ∩ cin 3883   ⊆ wss 3884  𝒫 cpw 4500  {cpr 4530  ⟨cop 4534  ∪ cuni 4803   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   E cep 5432   Se wse 5480   We wwe 5481   × cxp 5521  ◡ccnv 5522  dom cdm 5523  ran crn 5524   ↾ cres 5525   “ cima 5526  Ord word 6162  Oncon0 6163  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  1st c1st 7673  2nd c2nd 7674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-1st 7675  df-2nd 7676 This theorem is referenced by:  infxpenlem  9428
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