Theorem tfsconcatrn 43338

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 43333	. . 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 5905	. 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 6121	. . 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 3055	. . . . . 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 8513	. . . . . . . . . 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 776	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ∈ On)
13		simprr 772	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
14		onelon 6360	. . . . . . . . . 10 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
15	13, 14	sylan 580	. . . . . . . . 9 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
16		oaword1 8519	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝑑 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑑))
17	12, 15, 16	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑑))
18		oacl 8502	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
19	18	ad2antlr 727	. . . . . . . . . 10 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷) → (𝐶 +o 𝐷) ∈ On)
20		eloni 6345	. . . . . . . . . 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 6345	. . . . . . . . . 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 43254	. . . . . . . . 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 584	. . . . . . . 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 713	. . . . . . 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 43254	. . . . . . . . . . . 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 43322	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝑥 ∧ 𝑥 ∈ (𝐶 +o 𝐷))) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
38	28, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃𝑑 ∈ 𝐷 (𝐶 +o 𝑑) = 𝑥)
39		eqcom 2737	. . . . . . . . 9 ⊢ ((𝐶 +o 𝑑) = 𝑥 ↔ 𝑥 = (𝐶 +o 𝑑))
40	39	rexbii 3077	. . . . . . . 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 1215	. . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝐷 ∈ On)
48		onelon 6360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷 ∈ On ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
49	47, 48	sylan 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ On)
50		simp2 1137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑑 ∈ 𝐷 ∧ 𝑥 = (𝐶 +o 𝑑)) → 𝑑 ∈ 𝐷)
51	47, 50, 14	syl2anc 584	. . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . 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 8515	. . . . . . . . . . . . . . . . 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 4608	. . . . . . . . . . . . . . 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 772	. . . . . . . . . . 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 3144	. . . . . . . . 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 3454	. . . . . . . . . 10 ⊢ 𝑑 ∈ V
67		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → (𝐵‘𝑧) = (𝐵‘𝑑))
68	67	eqeq2d 2741	. . . . . . . . . 10 ⊢ (𝑧 = 𝑑 → (𝑦 = (𝐵‘𝑧) ↔ 𝑦 = (𝐵‘𝑑)))
69	66, 68	rexsn 4649	. . . . . . . . 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 1194	. . . . . . . . . 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 7398	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑑 → (𝐶 +o 𝑧) = (𝐶 +o 𝑑))
75	74	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑑 → (𝑥 = (𝐶 +o 𝑧) ↔ 𝑥 = (𝐶 +o 𝑑)))
76	75, 68	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑑 → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ (𝑥 = (𝐶 +o 𝑑) ∧ 𝑦 = (𝐵‘𝑑))))
77	76	rspcev 3591	. . . . . . . . . 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 836	. . . . . . . . 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 5371	. . . . . 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 2796	. . . 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 5921	. . . . 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 6923	. . . . 5 ⊢ (𝐵 Fn 𝐷 → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
87	86	ad2antlr 727	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran 𝐵 = {𝑦 ∣ ∃𝑑 ∈ 𝐷 𝑦 = (𝐵‘𝑑)})
88	83, 85, 87	3eqtr4d 2775	. . 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 4134	. 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 2769	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 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  {csn 4592  {copab 5172  dom cdm 5641  ran crn 5642  Ord word 6334  Oncon0 6335   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   +_o coa 8434

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441

This theorem is referenced by:  tfsconcatfo  43339  tfsconcat00  43343  tfsconcatrnss12  43345  tfsconcatrnss  43346