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Theorem fsplitfpar 8104
Description: Merge two functions with a common argument in parallel. Combination of fsplit 8103 and fpar 8102. (Contributed by AV, 3-Jan-2024.)
Hypotheses
Ref Expression
fsplitfpar.h 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
fsplitfpar.s 𝑆 = ((1st ↾ I ) ↾ 𝐴)
Assertion
Ref Expression
fsplitfpar ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺
Allowed substitution hints:   𝑆(𝑥)   𝐻(𝑥)

Proof of Theorem fsplitfpar
Dummy variables 𝑎 𝑝 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsplitfpar.s . . . . . . . . . 10 𝑆 = ((1st ↾ I ) ↾ 𝐴)
2 fsplit 8103 . . . . . . . . . . 11 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
32reseq1i 5978 . . . . . . . . . 10 ((1st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
41, 3eqtri 2761 . . . . . . . . 9 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
54fveq1i 6893 . . . . . . . 8 (𝑆𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎)
65a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑆𝑎) = (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎))
7 fvres 6911 . . . . . . . . 9 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8 eqidd 2734 . . . . . . . . . 10 (𝑎𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
109, 9opeq12d 4882 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
1110adantl 483 . . . . . . . . . 10 ((𝑎𝐴𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12 elex 3493 . . . . . . . . . 10 (𝑎𝐴𝑎 ∈ V)
13 opex 5465 . . . . . . . . . . 11 𝑎, 𝑎⟩ ∈ V
1413a1i 11 . . . . . . . . . 10 (𝑎𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
158, 11, 12, 14fvmptd 7006 . . . . . . . . 9 (𝑎𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
167, 15eqtrd 2773 . . . . . . . 8 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
1716adantl 483 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
186, 17eqtrd 2773 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑆𝑎) = ⟨𝑎, 𝑎⟩)
1918fveq2d 6896 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20 df-ov 7412 . . . . . 6 (𝑎𝐻𝑎) = (𝐻‘⟨𝑎, 𝑎⟩)
21 fsplitfpar.h . . . . . . . . 9 𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
2221fpar 8102 . . . . . . . 8 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
2322adantr 482 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
24 fveq2 6892 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
2524adantr 482 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹𝑥) = (𝐹𝑎))
26 fveq2 6892 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
2726adantl 483 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐺𝑦) = (𝐺𝑎))
2825, 27opeq12d 4882 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
2928adantl 483 . . . . . . 7 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) ∧ (𝑥 = 𝑎𝑦 = 𝑎)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
30 simpr 486 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
31 opex 5465 . . . . . . . 8 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3231a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V)
3323, 29, 30, 30, 32ovmpod 7560 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3420, 33eqtr3id 2787 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3519, 34eqtrd 2773 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
36 eqid 2733 . . . . . . . . . 10 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
3736fnmpt 6691 . . . . . . . . 9 (∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
3813a1i 11 . . . . . . . . 9 (𝑎 ∈ V → ⟨𝑎, 𝑎⟩ ∈ V)
3937, 38mprg 3068 . . . . . . . 8 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40 ssv 4007 . . . . . . . 8 𝐴 ⊆ V
41 fnssres 6674 . . . . . . . 8 (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
4239, 40, 41mp2an 691 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43 fsplit 8103 . . . . . . . . . 10 (1st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
4443reseq1i 5978 . . . . . . . . 9 ((1st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
451, 44eqtri 2761 . . . . . . . 8 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
4645fneq1i 6647 . . . . . . 7 (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
4742, 46mpbir 230 . . . . . 6 𝑆 Fn 𝐴
4847a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
49 fvco2 6989 . . . . 5 ((𝑆 Fn 𝐴𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
5048, 49sylan 581 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
51 fveq2 6892 . . . . . . 7 (𝑥 = 𝑎 → (𝐺𝑥) = (𝐺𝑎))
5224, 51opeq12d 4882 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
53 eqid 2733 . . . . . 6 (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
5452, 53, 31fvmpt 6999 . . . . 5 (𝑎𝐴 → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5554adantl 483 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5635, 50, 553eqtr4d 2783 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
5756ralrimiva 3147 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
58 opex 5465 . . . . . . . 8 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
5958a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝑥𝐴𝑦𝐴)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
6059ralrimivva 3201 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴𝑦𝐴 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
61 eqid 2733 . . . . . . 7 (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
6261fnmpo 8055 . . . . . 6 (∀𝑥𝐴𝑦𝐴 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) Fn (𝐴 × 𝐴))
6360, 62syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) Fn (𝐴 × 𝐴))
6422fneq1d 6643 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻 Fn (𝐴 × 𝐴) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) Fn (𝐴 × 𝐴)))
6563, 64mpbird 257 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 Fn (𝐴 × 𝐴))
6613a1i 11 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎 ∈ V) → ⟨𝑎, 𝑎⟩ ∈ V)
6766ralrimiva 3147 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V)
6867, 37syl 17 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
6968, 40, 41sylancl 587 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
7069, 46sylibr 233 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
7145rneqi 5937 . . . . . 6 ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72 mptima 6072 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73 df-ima 5690 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74 eqid 2733 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
7574rnmpt 5955 . . . . . . 7 ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7672, 73, 753eqtr3i 2769 . . . . . 6 ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7771, 76eqtri 2761 . . . . 5 ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78 elinel2 4197 . . . . . . . . 9 (𝑎 ∈ (V ∩ 𝐴) → 𝑎𝐴)
79 simpl 484 . . . . . . . . . . . 12 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎𝐴)
8079, 79opelxpd 5716 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81 eleq1 2822 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8281adantl 483 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8380, 82mpbird 257 . . . . . . . . . 10 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8483ex 414 . . . . . . . . 9 (𝑎𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8578, 84syl 17 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8685rexlimiv 3149 . . . . . . 7 (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
8786abssi 4068 . . . . . 6 {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴)
8887a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩} ⊆ (𝐴 × 𝐴))
8977, 88eqsstrid 4031 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90 fnco 6668 . . . 4 ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻𝑆) Fn 𝐴)
9165, 70, 89, 90syl3anc 1372 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) Fn 𝐴)
92 opex 5465 . . . . . 6 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
9392a1i 11 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9493ralrimiva 3147 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9553fnmpt 6691 . . . 4 (∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
9694, 95syl 17 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴)
97 eqfnfv 7033 . . 3 (((𝐻𝑆) Fn 𝐴 ∧ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9891, 96, 97syl2anc 585 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9957, 98mpbird 257 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  cin 3948  wss 3949  cop 4635  cmpt 5232   I cid 5574   × cxp 5675  ccnv 5676  ran crn 5678  cres 5679  cima 5680  ccom 5681   Fn wfn 6539  cfv 6544  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  offsplitfpar  8105
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