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Theorem fsplitfpar 8106
Description: Merge two functions with a common argument in parallel. Combination of fsplit 8105 and fpar 8104. (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 8105 . . . . . . . . . . 11 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
32reseq1i 5976 . . . . . . . . . 10 ((1st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
41, 3eqtri 2758 . . . . . . . . 9 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
54fveq1i 6891 . . . . . . . 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 6909 . . . . . . . . 9 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8 eqidd 2731 . . . . . . . . . 10 (𝑎𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
109, 9opeq12d 4880 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
1110adantl 480 . . . . . . . . . 10 ((𝑎𝐴𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12 elex 3491 . . . . . . . . . 10 (𝑎𝐴𝑎 ∈ V)
13 opex 5463 . . . . . . . . . . 11 𝑎, 𝑎⟩ ∈ V
1413a1i 11 . . . . . . . . . 10 (𝑎𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
158, 11, 12, 14fvmptd 7004 . . . . . . . . 9 (𝑎𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
167, 15eqtrd 2770 . . . . . . . 8 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
1716adantl 480 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
186, 17eqtrd 2770 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑆𝑎) = ⟨𝑎, 𝑎⟩)
1918fveq2d 6894 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20 df-ov 7414 . . . . . 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 8104 . . . . . . . 8 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
2322adantr 479 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
24 fveq2 6890 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
2524adantr 479 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹𝑥) = (𝐹𝑎))
26 fveq2 6890 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
2726adantl 480 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐺𝑦) = (𝐺𝑎))
2825, 27opeq12d 4880 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
2928adantl 480 . . . . . . 7 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) ∧ (𝑥 = 𝑎𝑦 = 𝑎)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
30 simpr 483 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
31 opex 5463 . . . . . . . 8 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3231a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V)
3323, 29, 30, 30, 32ovmpod 7562 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3420, 33eqtr3id 2784 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3519, 34eqtrd 2770 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
36 eqid 2730 . . . . . . . . . 10 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
3736fnmpt 6689 . . . . . . . . 9 (∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
3813a1i 11 . . . . . . . . 9 (𝑎 ∈ V → ⟨𝑎, 𝑎⟩ ∈ V)
3937, 38mprg 3065 . . . . . . . 8 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40 ssv 4005 . . . . . . . 8 𝐴 ⊆ V
41 fnssres 6672 . . . . . . . 8 (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
4239, 40, 41mp2an 688 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43 fsplit 8105 . . . . . . . . . 10 (1st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
4443reseq1i 5976 . . . . . . . . 9 ((1st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
451, 44eqtri 2758 . . . . . . . 8 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
4645fneq1i 6645 . . . . . . 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 6987 . . . . 5 ((𝑆 Fn 𝐴𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
5048, 49sylan 578 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
51 fveq2 6890 . . . . . . 7 (𝑥 = 𝑎 → (𝐺𝑥) = (𝐺𝑎))
5224, 51opeq12d 4880 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
53 eqid 2730 . . . . . 6 (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
5452, 53, 31fvmpt 6997 . . . . 5 (𝑎𝐴 → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5554adantl 480 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5635, 50, 553eqtr4d 2780 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
5756ralrimiva 3144 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
58 opex 5463 . . . . . . . 8 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
5958a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝑥𝐴𝑦𝐴)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
6059ralrimivva 3198 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴𝑦𝐴 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
61 eqid 2730 . . . . . . 7 (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
6261fnmpo 8057 . . . . . 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 6641 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻 Fn (𝐴 × 𝐴) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) Fn (𝐴 × 𝐴)))
6563, 64mpbird 256 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 Fn (𝐴 × 𝐴))
6613a1i 11 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎 ∈ V) → ⟨𝑎, 𝑎⟩ ∈ V)
6766ralrimiva 3144 . . . . . . 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 584 . . . . 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 5935 . . . . . 6 ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72 mptima 6070 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73 df-ima 5688 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74 eqid 2730 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
7574rnmpt 5953 . . . . . . 7 ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7672, 73, 753eqtr3i 2766 . . . . . 6 ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7771, 76eqtri 2758 . . . . 5 ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78 elinel2 4195 . . . . . . . . 9 (𝑎 ∈ (V ∩ 𝐴) → 𝑎𝐴)
79 simpl 481 . . . . . . . . . . . 12 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎𝐴)
8079, 79opelxpd 5714 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81 eleq1 2819 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8281adantl 480 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8380, 82mpbird 256 . . . . . . . . . 10 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8483ex 411 . . . . . . . . 9 (𝑎𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8578, 84syl 17 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8685rexlimiv 3146 . . . . . . 7 (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
8786abssi 4066 . . . . . 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 4029 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90 fnco 6666 . . . 4 ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻𝑆) Fn 𝐴)
9165, 70, 89, 90syl3anc 1369 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) Fn 𝐴)
92 opex 5463 . . . . . 6 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
9392a1i 11 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9493ralrimiva 3144 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9553fnmpt 6689 . . . 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 7031 . . 3 (((𝐻𝑆) Fn 𝐴 ∧ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9891, 96, 97syl2anc 582 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9957, 98mpbird 256 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  wral 3059  wrex 3068  Vcvv 3472  cin 3946  wss 3947  cop 4633  cmpt 5230   I cid 5572   × cxp 5673  ccnv 5674  ran crn 5676  cres 5677  cima 5678  ccom 5679   Fn wfn 6537  cfv 6542  (class class class)co 7411  cmpo 7413  1st c1st 7975  2nd c2nd 7976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978
This theorem is referenced by:  offsplitfpar  8107
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