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Theorem tfsconcatrev 43338
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 6749 . . . . . 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 6672 . . . . . . . 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 8572 . . . . . . . 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 2839 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom 𝐹 ∈ On)
9 fnfun 6669 . . . . . . 7 (𝐹 Fn (𝐶 +o 𝐷) → Fun 𝐹)
109adantr 480 . . . . . 6 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → Fun 𝐹)
11 funrnex 7977 . . . . . 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 8879 . . . 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 8589 . . . 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 8906 . . 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 8583 . . . . . . . 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 8879 . . 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 6693 . . . . 5 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹𝐶) Fn 𝐶)
31 fvex 6920 . . . . . . 7 (𝐹‘(𝐶 +o 𝑑)) ∈ V
32 eqid 2735 . . . . . . 7 (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))
3331, 32fnmpti 6712 . . . . . 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 43327 . . . . 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 6911 . . . . . . . . . . . . . . . . 17 (𝑑 = 𝑧 → (𝐹‘(𝐶 +o 𝑑)) = (𝐹‘(𝐶 +o 𝑧)))
41 fvex 6920 . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . 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 3153 . . . . . . . . . 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 6396 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
536, 52syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
54 eloni 6396 . . . . . . . . . . . . . . . . . . . 20 (𝐶 ∈ On → Ord 𝐶)
5554adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
56 ordeldif 43248 . . . . . . . . . . . . . . . . . . 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 43316 . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . 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 4141 . . . . . . . . . 10 (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) → 𝑥 ∈ (𝐶 +o 𝐷))
78 eqcom 2742 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
79 fnbrfvb 6960 . . . . . . . . . . 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 5214 . . . . . 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 6061 . . . . . 6 (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ 𝑥𝐹𝑦)}
8684, 85eqtr4di 2793 . . . . 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 4178 . . . 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 2775 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = ((𝐹𝐶) ∪ (𝐹 ↾ ((𝐶 +o 𝐷) ∖ 𝐶))))
89 resundi 6014 . . . 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 4488 . . . . . . 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 6000 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = (𝐹 ↾ (𝐶 +o 𝐷)))
95 fnresdm 6688 . . . . 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 2775 . . 3 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐹 ↾ (𝐶 ∪ ((𝐶 +o 𝐷) ∖ 𝐶))) = 𝐹)
9888, 90, 973eqtr2d 2781 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹)
99 dmres 6032 . . 3 dom (𝐹𝐶) = (𝐶 ∩ dom 𝐹)
10016, 5sseqtrrd 4037 . . . 4 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐶 ⊆ dom 𝐹)
101 dfss2 3981 . . . 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 2787 . 2 ((𝐹 Fn (𝐶 +o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → dom (𝐹𝐶) = 𝐶)
10431, 32dmmpti 6713 . . 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 2737 . . . 4 (𝑢 = (𝐹𝐶) → ((𝑢 + 𝑣) = 𝐹 ↔ ((𝐹𝐶) + 𝑣) = 𝐹))
108 dmeq 5917 . . . . 5 (𝑢 = (𝐹𝐶) → dom 𝑢 = dom (𝐹𝐶))
109108eqeq1d 2737 . . . 4 (𝑢 = (𝐹𝐶) → (dom 𝑢 = 𝐶 ↔ dom (𝐹𝐶) = 𝐶))
110107, 1093anbi12d 1436 . . 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 2737 . . . 4 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (((𝐹𝐶) + 𝑣) = 𝐹 ↔ ((𝐹𝐶) + (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑)))) = 𝐹))
113 dmeq 5917 . . . . 5 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → dom 𝑣 = dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))))
114113eqeq1d 2737 . . . 4 (𝑣 = (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) → (dom 𝑣 = 𝐷 ↔ dom (𝑑𝐷 ↦ (𝐹‘(𝐶 +o 𝑑))) = 𝐷))
115112, 1143anbi13d 1437 . . 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 1381 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 1086   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963   class class class wbr 5148  {copab 5210  cmpt 5231  dom cdm 5689  ran crn 5690  cres 5691  Ord word 6385  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433   +o coa 8502  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-map 8867
This theorem is referenced by: (None)
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