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Theorem fsplitfpar 7806
 Description: Merge two functions with a common argument in parallel. Combination of fsplit 7804 and fpar 7803. (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 7804 . . . . . . . . . . 11 (1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
32reseq1i 5842 . . . . . . . . . 10 ((1st ↾ I ) ↾ 𝐴) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
41, 3eqtri 2842 . . . . . . . . 9 𝑆 = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)
54fveq1i 6664 . . . . . . . 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 6682 . . . . . . . . 9 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎))
8 eqidd 2820 . . . . . . . . . 10 (𝑎𝐴 → (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩))
9 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
109, 9opeq12d 4803 . . . . . . . . . . 11 (𝑥 = 𝑎 → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
1110adantl 484 . . . . . . . . . 10 ((𝑎𝐴𝑥 = 𝑎) → ⟨𝑥, 𝑥⟩ = ⟨𝑎, 𝑎⟩)
12 elex 3511 . . . . . . . . . 10 (𝑎𝐴𝑎 ∈ V)
13 opex 5347 . . . . . . . . . . 11 𝑎, 𝑎⟩ ∈ V
1413a1i 11 . . . . . . . . . 10 (𝑎𝐴 → ⟨𝑎, 𝑎⟩ ∈ V)
158, 11, 12, 14fvmptd 6768 . . . . . . . . 9 (𝑎𝐴 → ((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)‘𝑎) = ⟨𝑎, 𝑎⟩)
167, 15eqtrd 2854 . . . . . . . 8 (𝑎𝐴 → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
1716adantl 484 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (((𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩) ↾ 𝐴)‘𝑎) = ⟨𝑎, 𝑎⟩)
186, 17eqtrd 2854 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑆𝑎) = ⟨𝑎, 𝑎⟩)
1918fveq2d 6667 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = (𝐻‘⟨𝑎, 𝑎⟩))
20 df-ov 7151 . . . . . 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 7803 . . . . . . . 8 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
2322adantr 483 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝐻 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩))
24 fveq2 6663 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
2524adantr 483 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐹𝑥) = (𝐹𝑎))
26 fveq2 6663 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝐺𝑦) = (𝐺𝑎))
2726adantl 484 . . . . . . . . 9 ((𝑥 = 𝑎𝑦 = 𝑎) → (𝐺𝑦) = (𝐺𝑎))
2825, 27opeq12d 4803 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑎) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
2928adantl 484 . . . . . . 7 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) ∧ (𝑥 = 𝑎𝑦 = 𝑎)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
30 simpr 487 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → 𝑎𝐴)
31 opex 5347 . . . . . . . 8 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3231a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V)
3323, 29, 30, 30, 32ovmpod 7294 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝑎𝐻𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3420, 33syl5eqr 2868 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘⟨𝑎, 𝑎⟩) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
3519, 34eqtrd 2854 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → (𝐻‘(𝑆𝑎)) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
36 eqid 2819 . . . . . . . . . 10 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
3736fnmpt 6481 . . . . . . . . 9 (∀𝑎 ∈ V ⟨𝑎, 𝑎⟩ ∈ V → (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V)
3813a1i 11 . . . . . . . . 9 (𝑎 ∈ V → ⟨𝑎, 𝑎⟩ ∈ V)
3937, 38mprg 3150 . . . . . . . 8 (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V
40 ssv 3989 . . . . . . . 8 𝐴 ⊆ V
41 fnssres 6463 . . . . . . . 8 (((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) Fn V ∧ 𝐴 ⊆ V) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
4239, 40, 41mp2an 690 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴
43 fsplit 7804 . . . . . . . . . 10 (1st ↾ I ) = (𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩)
4443reseq1i 5842 . . . . . . . . 9 ((1st ↾ I ) ↾ 𝐴) = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
451, 44eqtri 2842 . . . . . . . 8 𝑆 = ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
4645fneq1i 6443 . . . . . . 7 (𝑆 Fn 𝐴 ↔ ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
4742, 46mpbir 233 . . . . . 6 𝑆 Fn 𝐴
4847a1i 11 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
49 fvco2 6751 . . . . 5 ((𝑆 Fn 𝐴𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
5048, 49sylan 582 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = (𝐻‘(𝑆𝑎)))
51 fveq2 6663 . . . . . . 7 (𝑥 = 𝑎 → (𝐺𝑥) = (𝐺𝑎))
5224, 51opeq12d 4803 . . . . . 6 (𝑥 = 𝑎 → ⟨(𝐹𝑥), (𝐺𝑥)⟩ = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
53 eqid 2819 . . . . . 6 (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)
5452, 53, 31fvmpt 6761 . . . . 5 (𝑎𝐴 → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5554adantl 484 . . . 4 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎) = ⟨(𝐹𝑎), (𝐺𝑎)⟩)
5635, 50, 553eqtr4d 2864 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎𝐴) → ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
5756ralrimiva 3180 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎))
58 opex 5347 . . . . . . . 8 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V
5958a1i 11 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝑥𝐴𝑦𝐴)) → ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
6059ralrimivva 3189 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴𝑦𝐴 ⟨(𝐹𝑥), (𝐺𝑦)⟩ ∈ V)
61 eqid 2819 . . . . . . 7 (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩)
6261fnmpo 7759 . . . . . 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 6439 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻 Fn (𝐴 × 𝐴) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑦)⟩) Fn (𝐴 × 𝐴)))
6563, 64mpbird 259 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐻 Fn (𝐴 × 𝐴))
6613a1i 11 . . . . . . . 8 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑎 ∈ V) → ⟨𝑎, 𝑎⟩ ∈ V)
6766ralrimiva 3180 . . . . . . 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 588 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) Fn 𝐴)
7069, 46sylibr 236 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑆 Fn 𝐴)
7145rneqi 5800 . . . . . 6 ran 𝑆 = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
72 mptima 5934 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
73 df-ima 5561 . . . . . . 7 ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) “ 𝐴) = ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴)
74 eqid 2819 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩)
7574rnmpt 5820 . . . . . . 7 ran (𝑎 ∈ (V ∩ 𝐴) ↦ ⟨𝑎, 𝑎⟩) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7672, 73, 753eqtr3i 2850 . . . . . 6 ran ((𝑎 ∈ V ↦ ⟨𝑎, 𝑎⟩) ↾ 𝐴) = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
7771, 76eqtri 2842 . . . . 5 ran 𝑆 = {𝑝 ∣ ∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩}
78 elinel2 4171 . . . . . . . . 9 (𝑎 ∈ (V ∩ 𝐴) → 𝑎𝐴)
79 simpl 485 . . . . . . . . . . . 12 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑎𝐴)
8079, 79opelxpd 5586 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴))
81 eleq1 2898 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑎⟩ → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8281adantl 484 . . . . . . . . . . 11 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → (𝑝 ∈ (𝐴 × 𝐴) ↔ ⟨𝑎, 𝑎⟩ ∈ (𝐴 × 𝐴)))
8380, 82mpbird 259 . . . . . . . . . 10 ((𝑎𝐴𝑝 = ⟨𝑎, 𝑎⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8483ex 415 . . . . . . . . 9 (𝑎𝐴 → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8578, 84syl 17 . . . . . . . 8 (𝑎 ∈ (V ∩ 𝐴) → (𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴)))
8685rexlimiv 3278 . . . . . . 7 (∃𝑎 ∈ (V ∩ 𝐴)𝑝 = ⟨𝑎, 𝑎⟩ → 𝑝 ∈ (𝐴 × 𝐴))
8786abssi 4044 . . . . . 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 4013 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ran 𝑆 ⊆ (𝐴 × 𝐴))
90 fnco 6458 . . . 4 ((𝐻 Fn (𝐴 × 𝐴) ∧ 𝑆 Fn 𝐴 ∧ ran 𝑆 ⊆ (𝐴 × 𝐴)) → (𝐻𝑆) Fn 𝐴)
9165, 70, 89, 90syl3anc 1365 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) Fn 𝐴)
92 opex 5347 . . . . . 6 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V
9392a1i 11 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9493ralrimiva 3180 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ∀𝑥𝐴 ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ V)
9553fnmpt 6481 . . . 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 6795 . . 3 (((𝐻𝑆) Fn 𝐴 ∧ (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9891, 96, 97syl2anc 586 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩) ↔ ∀𝑎𝐴 ((𝐻𝑆)‘𝑎) = ((𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)‘𝑎)))
9957, 98mpbird 259 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107  {cab 2797  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ⟨cop 4565   ↦ cmpt 5137   I cid 5452   × cxp 5546  ◡ccnv 5547  ran crn 5549   ↾ cres 5550   “ cima 5551   ∘ ccom 5552   Fn wfn 6343  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150  1st c1st 7679  2nd c2nd 7680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  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-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682 This theorem is referenced by:  offsplitfpar  7807
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