Theorem tfsconcatrn 43773

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 43768	. . 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 5885	. 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 6101	. . 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 3063	. . . . . 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		pm3.22 459	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
8	7	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
9		oaordi 8472	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑑 ∈ 𝐷 → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷)))
11	10	imp 406	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ (𝐶 +o 𝐷))
12		simplrl 777	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ∈ On)
13		simprr 773	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
14		onelon 6340	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
15	13, 14	sylan 581	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
16		oaword1 8478	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑑 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑑))
17	12, 15, 16	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑑))
18		oacl 8461	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
19	18	ad2antlr 728	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝐷) ∈ On)
20		eloni 6325	. . . . . . . . . 10 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → Ord (𝐶 +o 𝐷))
22		eloni 6325	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → Ord 𝐶)
23	12, 22	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → Ord 𝐶)
24		ordeldif 43689	. . . . . . . . 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 585	. . . . . . . 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 714	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝑑) ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
27		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
28	27	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ∈ On ∧ 𝐷 ∈ On))
29	18, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
30	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
31	29, 30	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
33		ordeldif 43689	. . . . . . . . . . . 12 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝑥 ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ 𝑥)))
34	32, 33	syl 17	. . . . . . . . . . 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 476	. . . . . . . . . 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 461	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷)))
37		oawordex2 43757	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
38	28, 36, 37	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
39		eqcom 2744	. . . . . . . . 9 ⊢ ((𝐶 +o 𝑑) = 𝑥 ↔ 𝑥 = (𝐶 +o 𝑑))
40	39	rexbii 3085	. . . . . . . 8 ⊢ (∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥 ↔ ∃𝑑 ∈ 𝐷 𝑥 = (𝐶 +o 𝑑))
41	38, 40	sylib 218	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃𝑑 ∈ 𝐷 𝑥 = (𝐶 +o 𝑑))
42		simpr 484	. . . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝐶 ∈ On)
46	45	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝐶 ∈ On)
47		simp1rr 1241	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝐷 ∈ On)
48		onelon 6340	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ On ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
49	47, 48	sylan 581	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
50		simp2 1138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝑑 ∈ 𝐷)
51	47, 50, 14	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝑑 ∈ On)
52	51	adantr 480	. . . . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 8474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ On ∧ 𝑧 ∈ On ∧ 𝑑 ∈ On) → ((𝐶 +o 𝑧) = (𝐶 +o 𝑑) ↔ 𝑧 = 𝑑))
56	54, 55	syl 17	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 4584	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑑} ↔ 𝑧 = 𝑑)
59	57, 58	sylibr 234	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 453	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) → 𝑦 = (𝐵‘𝑧))
64	62, 63	jca2 513	. . . . . . . . . 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 3148	. . . . . . . . 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 3434	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
67		fveq2 6832	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → (𝐵‘𝑧) = (𝐵‘𝑑))
68	67	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑧 = 𝑑 → (𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑑)))
69	66, 68	rexsn 4627	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑑}𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑑))
70	65, 69	imbitrdi 251	. . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑦 = (𝐵‘𝑑)) → 𝑑 ∈ 𝐷)
72		simpl3 1195	. . . . . . . . . 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 484	. . . . . . . . . 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 7366	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑑 → (𝐶 +o 𝑧) = (𝐶 +o 𝑑))
75	74	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑑 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝑥 = (𝐶 +o 𝑑)))
76	75, 68	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = (𝐶 +o 𝑑) ∧ 𝑦 = (𝐵‘𝑑))))
77	76	rspcev 3565	. . . . . . . . . 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 837	. . . . . . . . 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 412	. . . . . . . 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 212	. . . . . . 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 5348	. . . . . 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 285	. . . . 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 2803	. . . 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 5901	. . . . 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 6891	. . . . 5 ⊢ (𝐵 Fn 𝐷 → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
87	86	ad2antlr 728	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
88	83, 85, 87	3eqtr4d 2782	. . 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 4109	. 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 2776	1 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568  {copab 5148  dom cdm 5622  ran crn 5623  Ord word 6314  Oncon0 6315   Fn wfn 6485  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360   +_o coa 8393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-oadd 8400

This theorem is referenced by:  tfsconcatfo  43774  tfsconcat00  43778  tfsconcatrnss12  43780  tfsconcatrnss  43781