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Theorem tfsconcatrev 43361
Description: If the domain of a transfinite sequence is an ordinal sum, the sequence can be decomposed into two sequences with domains corresponding to the addends. Theorem 2 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
Hypothesis
Ref Expression
tfsconcat.op + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
Assertion
Ref Expression
tfsconcatrev ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹m 𝐶)∃𝑣 ∈ (ran 𝐹m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
Distinct variable groups:   𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧,𝐶   𝐷,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧   𝐹,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧   𝑢, + ,𝑣
Allowed substitution hints:   + (𝑥,𝑦,𝑧,𝑎,𝑏)

Proof of Theorem tfsconcatrev
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 dffn3 6748 . . . . . 6 (𝐹 Fn (𝐶 +o 𝐷) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
21biimpi 216 . . . . 5 (𝐹 Fn (𝐶 +o 𝐷) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
32adantr 480 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹)
4 fndm 6671 . . . . . . . 8 (𝐹 Fn (𝐶 +o 𝐷) → dom 𝐹 = (𝐶 +o 𝐷))
54adantr 480 . . . . . . 7 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 = (𝐶 +o 𝐷))
6 oacl 8573 . . . . . . . 8 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
76adantl 481 . . . . . . 7 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 +o 𝐷) ∈ On)
85, 7eqeltrd 2841 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 ∈ On)
9 fnfun 6668 . . . . . . 7 (𝐹 Fn (𝐶 +o 𝐷) → Fun 𝐹)
109adantr 480 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → Fun 𝐹)
11 funrnex 7978 . . . . . 6 (dom 𝐹 ∈ On → (Fun 𝐹 → ran 𝐹 ∈ V))
128, 10, 11sylc 65 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐹 ∈ V)
1312, 7elmapd 8880 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ∈ (ran 𝐹m (𝐶 +o 𝐷)) ↔ 𝐹:(𝐶 +o 𝐷)⟶ran 𝐹))
143, 13mpbird 257 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 ∈ (ran 𝐹m (𝐶 +o 𝐷)))
15 oaword1 8590 . . . 4 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝐷))
1615adantl 481 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ (𝐶 +o 𝐷))
17 elmapssres 8907 . . 3 ((𝐹 ∈ (ran 𝐹m (𝐶 +o 𝐷)) ∧ 𝐶 ⊆ (𝐶 +o 𝐷)) → (𝐹𝐶) ∈ (ran 𝐹m 𝐶))
1814, 16, 17syl2anc 584 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹𝐶) ∈ (ran 𝐹m 𝐶))
19 simpl 482 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐹 Fn (𝐶 +o 𝐷))
20 oaordi 8584 . . . . . . . 8 ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
2120ancoms 458 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑑𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
2221adantl 481 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
2322imp 406 . . . . 5 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
24 fnfvelrn 7100 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
2519, 23, 24syl2an2r 685 . . . 4 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑𝐷) → (𝐹‘(𝐶 +o 𝑑)) ∈ ran 𝐹)
2625fmpttd 7135 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹)
27 simprr 773 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
2812, 27elmapd 8880 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹m 𝐷) ↔ (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))):𝐷⟶ran 𝐹))
2926, 28mpbird 257 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹m 𝐷))
3019, 16fnssresd 6692 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹𝐶) Fn 𝐶)
31 fvex 6919 . . . . . . 7 (𝐹‘(𝐶 +o 𝑑)) ∈ V
32 eqid 2737 . . . . . . 7 (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))
3331, 32fnmpti 6711 . . . . . 6 (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷
3433a1i 11 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷)
35 simpr 484 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
36 tfsconcat.op . . . . . 6 + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏𝑧)))}))
3736tfsconcatun 43350 . . . . 5 ((((𝐹𝐶) Fn 𝐶 ∧ (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}))
3830, 34, 35, 37syl21anc 838 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}))
39 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (𝑑 = 𝑧 → (𝐶 +o 𝑑) = (𝐶 +o 𝑧))
4039fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑧 → (𝐹‘(𝐶 +o 𝑑)) = (𝐹‘(𝐶 +o 𝑧)))
41 fvex 6919 . . . . . . . . . . . . . . . . 17 (𝐹‘(𝐶 +o 𝑧)) ∈ V
4240, 32, 41fvmpt 7016 . . . . . . . . . . . . . . . 16 (𝑧𝐷 → ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
4342ad2antlr 727 . . . . . . . . . . . . . . 15 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
44 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑥 = (𝐶 +o 𝑧) → (𝐹𝑥) = (𝐹‘(𝐶 +o 𝑧)))
4544adantl 481 . . . . . . . . . . . . . . 15 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝐹𝑥) = (𝐹‘(𝐶 +o 𝑧)))
4643, 45eqtr4d 2780 . . . . . . . . . . . . . 14 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹𝑥))
4746eqeq2d 2748 . . . . . . . . . . . . 13 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) ↔ 𝑦 = (𝐹𝑥)))
4847biimpd 229 . . . . . . . . . . . 12 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) → 𝑦 = (𝐹𝑥)))
4948expimpd 453 . . . . . . . . . . 11 ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑧𝐷) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) → 𝑦 = (𝐹𝑥)))
5049rexlimdva 3155 . . . . . . . . . 10 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) → 𝑦 = (𝐹𝑥)))
51 simplr 769 . . . . . . . . . . . . . . 15 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
52 eloni 6394 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
536, 52syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
54 eloni 6394 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ∈ On → Ord 𝐶)
5554adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
56 ordeldif 43271 . . . . . . . . . . . . . . . . . . 19 ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶𝑥)))
5753, 55, 56syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶𝑥)))
5857adantl 481 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶𝑥)))
5958biimpa 476 . . . . . . . . . . . . . . . 16 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶𝑥))
6059ancomd 461 . . . . . . . . . . . . . . 15 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶𝑥𝑥 ∈ (𝐶 +o 𝐷)))
6151, 60jca 511 . . . . . . . . . . . . . 14 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶𝑥𝑥 ∈ (𝐶 +o 𝐷))))
6261adantr 480 . . . . . . . . . . . . 13 ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) → ((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶𝑥𝑥 ∈ (𝐶 +o 𝐷))))
63 oawordex2 43339 . . . . . . . . . . . . 13 (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶𝑥𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑧𝐷 (𝐶 +o 𝑧) = 𝑥)
6462, 63syl 17 . . . . . . . . . . . 12 ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) → ∃𝑧𝐷 (𝐶 +o 𝑧) = 𝑥)
65 simpr 484 . . . . . . . . . . . . . . . 16 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝐶 +o 𝑧) = 𝑥)
6665eqcomd 2743 . . . . . . . . . . . . . . 15 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑥 = (𝐶 +o 𝑧))
6765fveq2d 6910 . . . . . . . . . . . . . . . 16 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝐹‘(𝐶 +o 𝑧)) = (𝐹𝑥))
6842ad2antlr 727 . . . . . . . . . . . . . . . 16 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧) = (𝐹‘(𝐶 +o 𝑧)))
69 simpllr 776 . . . . . . . . . . . . . . . 16 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑦 = (𝐹𝑥))
7067, 68, 693eqtr4rd 2788 . . . . . . . . . . . . . . 15 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))
7166, 70jca 511 . . . . . . . . . . . . . 14 ((((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) ∧ (𝐶 +o 𝑧) = 𝑥) → (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))
7271ex 412 . . . . . . . . . . . . 13 (((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) ∧ 𝑧𝐷) → ((𝐶 +o 𝑧) = 𝑥 → (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
7372reximdva 3168 . . . . . . . . . . . 12 ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) → (∃𝑧𝐷 (𝐶 +o 𝑧) = 𝑥 → ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
7464, 73mpd 15 . . . . . . . . . . 11 ((((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) ∧ 𝑦 = (𝐹𝑥)) → ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))
7574ex 412 . . . . . . . . . 10 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹𝑥) → ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))))
7650, 75impbid 212 . . . . . . . . 9 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) ↔ 𝑦 = (𝐹𝑥)))
77 eldifi 4131 . . . . . . . . . 10 (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) → 𝑥 ∈ (𝐶 +o 𝐷))
78 eqcom 2744 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
79 fnbrfvb 6959 . . . . . . . . . . 11 ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
8078, 79bitrid 283 . . . . . . . . . 10 ((𝐹 Fn (𝐶 +o 𝐷) ∧ 𝑥 ∈ (𝐶 +o 𝐷)) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
8119, 77, 80syl2an 596 . . . . . . . . 9 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
8276, 81bitrd 279 . . . . . . . 8 (((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)) ↔ 𝑥𝐹𝑦))
8382pm5.32da 579 . . . . . . 7 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧))) ↔ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)))
8483opabbidv 5209 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)})
85 dfres2 6059 . . . . . 6 (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)}
8684, 85eqtr4di 2795 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))} = (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)))
8786uneq2d 4168 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = ((𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))‘𝑧)))}) = ((𝐹𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
8838, 87eqtrd 2777 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
89 resundi 6011 . . . 4 (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = ((𝐹𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)))
9089a1i 11 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = ((𝐹𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
91 undif 4482 . . . . . . 7 (𝐶 ⊆ (𝐶 +o 𝐷) ↔ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
9215, 91sylib 218 . . . . . 6 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
9392adantl 481 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶)) = (𝐶 +o 𝐷))
9493reseq2d 5997 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = (𝐹 ↾ (𝐶 +o 𝐷)))
95 fnresdm 6687 . . . . 5 (𝐹 Fn (𝐶 +o 𝐷) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
9695adantr 480 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 +o 𝐷)) = 𝐹)
9794, 96eqtrd 2777 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = 𝐹)
9888, 90, 973eqtr2d 2783 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹)
99 dmres 6030 . . 3 dom (𝐹𝐶) = (𝐶 ∩ dom 𝐹)
10016, 5sseqtrrd 4021 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ dom 𝐹)
101 dfss2 3969 . . . 4 (𝐶 ⊆ dom 𝐹 ↔ (𝐶 ∩ dom 𝐹) = 𝐶)
102100, 101sylib 218 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∩ dom 𝐹) = 𝐶)
10399, 102eqtrid 2789 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝐹𝐶) = 𝐶)
10431, 32dmmpti 6712 . . 3 dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷
105104a1i 11 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)
106 oveq1 7438 . . . . 5 (𝑢 = (𝐹𝐶) → (𝑢 + 𝑣) = ((𝐹𝐶) + 𝑣))
107106eqeq1d 2739 . . . 4 (𝑢 = (𝐹𝐶) → ((𝑢 + 𝑣) = 𝐹 ↔ ((𝐹𝐶) + 𝑣) = 𝐹))
108 dmeq 5914 . . . . 5 (𝑢 = (𝐹𝐶) → dom 𝑢 = dom (𝐹𝐶))
109108eqeq1d 2739 . . . 4 (𝑢 = (𝐹𝐶) → (dom 𝑢 = 𝐶 ↔ dom (𝐹𝐶) = 𝐶))
110107, 1093anbi12d 1439 . . 3 (𝑢 = (𝐹𝐶) → (((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷) ↔ (((𝐹𝐶) + 𝑣) = 𝐹 ∧ dom (𝐹𝐶) = 𝐶 ∧ dom 𝑣 = 𝐷)))
111 oveq2 7439 . . . . 5 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((𝐹𝐶) + 𝑣) = ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))))
112111eqeq1d 2739 . . . 4 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (((𝐹𝐶) + 𝑣) = 𝐹 ↔ ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹))
113 dmeq 5914 . . . . 5 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → dom 𝑣 = dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))))
114113eqeq1d 2739 . . . 4 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (dom 𝑣 = 𝐷 ↔ dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷))
115112, 1143anbi13d 1440 . . 3 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → ((((𝐹𝐶) + 𝑣) = 𝐹 ∧ dom (𝐹𝐶) = 𝐶 ∧ dom 𝑣 = 𝐷) ↔ (((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹 ∧ dom (𝐹𝐶) = 𝐶 ∧ dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)))
116110, 115rspc2ev 3635 . 2 (((𝐹𝐶) ∈ (ran 𝐹m 𝐶) ∧ (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) ∈ (ran 𝐹m 𝐷) ∧ (((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹 ∧ dom (𝐹𝐶) = 𝐶 ∧ dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷)) → ∃𝑢 ∈ (ran 𝐹m 𝐶)∃𝑣 ∈ (ran 𝐹m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
11718, 29, 98, 103, 105, 116syl113anc 1384 1 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹m 𝐶)∃𝑣 ∈ (ran 𝐹m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951   class class class wbr 5143  {copab 5205  cmpt 5225  dom cdm 5685  ran crn 5686  cres 5687  Ord word 6383  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433   +o coa 8503  m cmap 8866
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-map 8868
This theorem is referenced by: (None)
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