Theorem tfsconcatrn 44536

Description: The range of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-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 = (𝑏‘𝑧)))}))

Assertion

Ref	Expression
tfsconcatrn	⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑥,𝑦,𝑧   𝐶,𝑎,𝑏,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑥,𝑦,𝑧

Allowed substitution hints:   + (𝑥, 𝑦, 𝑧, 𝑎, 𝑏)

Proof of Theorem tfsconcatrn

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfsconcat.op	. . . 4 ⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))}))
2	1	tfsconcatun 44531	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
3	2	rneqd 5927	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = ran (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
4		rnun 6148	. . 3 ⊢ ran (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = (ran 𝐴 ∪ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})
5	4	a1i 11	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = (ran 𝐴 ∪ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
6		df-rex 3091	. . . . . 6 ⊢ (∃𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑥(𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
7		pm3.22 465	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
8	7	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
9		oaordi 8582	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
11	10	imp 412	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
12		simplrl 789	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ∈ On)
13		simprr 785	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
14		onelon 6405	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
15	13, 14	sylan 592	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
16		oaword1 8588	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑑 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑑))
17	12, 15, 16	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑑))
18		oacl 8571	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
19	18	ad2antlr 740	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝐷) ∈ On)
20		eloni 6390	. . . . . . . . . 10 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
21	19, 20	syl 18	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → Ord (𝐶 +o 𝐷))
22		eloni 6390	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → Ord 𝐶)
23	12, 22	syl 18	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → Ord 𝐶)
24		ordeldif 44452	. . . . . . . . 9 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → ((𝐶 +o 𝑑) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑑))))
25	21, 23, 24	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → ((𝐶 +o 𝑑) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑑))))
26	11, 17, 25	mpbir2and 726	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
27		simpr 490	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
28	27	adantr 486	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
29	18, 20	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
30	22	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
31	29, 30	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
32	31	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
33		ordeldif 44452	. . . . . . . . . . . 12 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
34	32, 33	syl 18	. . . . . . . . . . 11 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
35	34	biimpa 482	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥))
36	35	ancomd 467	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷)))
37		oawordex2 44520	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
38	28, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
39		eqcom 2770	. . . . . . . . 9 ⊢ ((𝐶 +o 𝑑) = 𝑥 ↔ 𝑥 = (𝐶 +o 𝑑))
40	39	rexbii 3113	. . . . . . . 8 ⊢ (∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥 ↔ ∃𝑑 ∈ 𝐷 𝑥 = (𝐶 +o 𝑑))
41	38, 40	sylib 221	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃𝑑 ∈ 𝐷 𝑥 = (𝐶 +o 𝑑))
42		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → 𝑥 = (𝐶 +o 𝑧))
43		simpll3 1233	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → 𝑥 = (𝐶 +o 𝑑))
44	42, 43	eqtr3d 2800	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝐶 +o 𝑧) = (𝐶 +o 𝑑))
45		simp1rl 1257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝐶 ∈ On)
46	45	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝐶 ∈ On)
47		simp1rr 1258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝐷 ∈ On)
48		onelon 6405	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ On ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
49	47, 48	sylan 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
50		simp2 1155	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝑑 ∈ 𝐷)
51	47, 50, 14	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝑑 ∈ On)
52	51	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝑑 ∈ On)
53	46, 49, 52	3jca 1146	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → (𝐶 ∈ On ∧ 𝑧 ∈ On ∧ 𝑑 ∈ On))
54	53	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → (𝐶 ∈ On ∧ 𝑧 ∈ On ∧ 𝑑 ∈ On))
55		oacan 8584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On ∧ 𝑑 ∈ On) → ((𝐶 +o 𝑧) = (𝐶 +o 𝑑) ↔ 𝑧 = 𝑑))
56	54, 55	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → ((𝐶 +o 𝑧) = (𝐶 +o 𝑑) ↔ 𝑧 = 𝑑))
57	44, 56	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → 𝑧 = 𝑑)
58		velsn 4600	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑑} ↔ 𝑧 = 𝑑)
59	57, 58	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 = (𝐶 +o 𝑧)) → 𝑧 ∈ {𝑑})
60	59	ex 418	. . . . . . . . . . . . 13 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → (𝑥 = (𝐶 +o 𝑧) → 𝑧 ∈ {𝑑}))
61	60	adantrd 497	. . . . . . . . . . . 12 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) → 𝑧 ∈ {𝑑}))
62	61	expimpd 459	. . . . . . . . . . 11 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → ((𝑧 ∈ 𝐷 ∧ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) → 𝑧 ∈ {𝑑}))
63		simprr 785	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) → 𝑦 = (𝐵‘𝑧))
64	62, 63	jca2 523	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → ((𝑧 ∈ 𝐷 ∧ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) → (𝑧 ∈ {𝑑} ∧ 𝑦 = (𝐵‘𝑧))))
65	64	reximdv2 3176	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) → ∃𝑧 ∈ {𝑑}𝑦 = (𝐵‘𝑧)))
66		vex 3458	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
67		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → (𝐵‘𝑧) = (𝐵‘𝑑))
68	67	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑧 = 𝑑 → (𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑑)))
69	66, 68	rexsn 4643	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑑}𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑑))
70	65, 69	imbitrdi 254	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) → 𝑦 = (𝐵‘𝑑)))
71	50	adantr 486	. . . . . . . . . 10 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑦 = (𝐵‘𝑑)) → 𝑑 ∈ 𝐷)
72		simpl3 1212	. . . . . . . . . 10 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑦 = (𝐵‘𝑑)) → 𝑥 = (𝐶 +o 𝑑))
73		simpr 490	. . . . . . . . . 10 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑦 = (𝐵‘𝑑)) → 𝑦 = (𝐵‘𝑑))
74		oveq2 7451	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑑 → (𝐶 +o 𝑧) = (𝐶 +o 𝑑))
75	74	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑑 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝑥 = (𝐶 +o 𝑑)))
76	75, 68	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = (𝐶 +o 𝑑) ∧ 𝑦 = (𝐵‘𝑑))))
77	76	rspcev 3580	. . . . . . . . . 10 ⊢ ((𝑑 ∈ 𝐷 ∧ (𝑥 = (𝐶 +o 𝑑) ∧ 𝑦 = (𝐵‘𝑑))) → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
78	71, 72, 73, 77	syl12anc 850	. . . . . . . . 9 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑦 = (𝐵‘𝑑)) → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
79	78	ex 418	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → (𝑦 = (𝐵‘𝑑) → ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
80	70, 79	impbid 215	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ 𝑦 = (𝐵‘𝑑)))
81	26, 41, 80	rexxfrd2 5379	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (∃𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)))
82	6, 81	bitr3id 288	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (∃𝑥(𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)))
83	82	abbidv 2829	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {𝑦 ∣ ∃𝑥(𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
84		rnopab 5943	. . . . 5 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}
85	84	a1i 11	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {𝑦 ∣ ∃𝑥(𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})
86		fnrnfv 6967	. . . . 5 ⊢ (𝐵 Fn 𝐷 → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
87	86	ad2antlr 740	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
88	83, 85, 87	3eqtr4d 2808	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = ran 𝐵)
89	88	uneq2d 4170	. 2 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran 𝐴 ∪ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}) = (ran 𝐴 ∪ ran 𝐵))
90	3, 5, 89	3eqtrd 2802	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  {csn 4588  {copab 5170  dom cdm 5655  ran crn 5656  Ord word 6379  Oncon0 6380   Fn wfn 6553  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   +_o coa 8501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-oadd 8508

This theorem is used by:  tfsconcatfo  44537  tfsconcat00  44541  tfsconcatrnss12  44543  tfsconcatrnss  44544