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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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