Theorem tfsconcatfv2 43353

Description: A latter value of the concatenation of two transfinite series. (Contributed by RP, 23-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
tfsconcatfv2	⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))

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   𝑥,𝑋,𝑦,𝑧

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

Proof of Theorem tfsconcatfv2

Step	Hyp	Ref	Expression
1		tfsconcat.op	. . . . 5 ⊢ + = (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 43350	. . . 4 ⊢ (((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	fveq1d 6908	. . 3 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})‘(𝐶 +o 𝑋)))
4	3	adantr 480	. 2 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})‘(𝐶 +o 𝑋)))
5		simplll 775	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐴 Fn 𝐶)
6		simplrl 777	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
7		simplrr 778	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
8		simpr 484	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
9		tfsconcatlem 43349	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
10	6, 7, 8, 9	syl3anc 1373	. . . . . 6 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → ∃!𝑦∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
11	10	ralrimiva 3146	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))
12		eqid 2737	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}
13	12	fnopabg 6705	. . . . 5 ⊢ (∀𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)∃!𝑦∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
14	11, 13	sylib 218	. . . 4 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
15	14	adantr 480	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶))
16		disjdif 4472	. . . 4 ⊢ (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅
17	16	a1i 11	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (𝐶 ∩ ((𝐶 +o 𝐷) ∖ 𝐶)) = ∅)
18		pm3.22 459	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
19	18	adantl 481	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
20		oaordi 8584	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ∈ 𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
21	19, 20	syl 17	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑋 ∈ 𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
22	21	imp 406	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷))
23		simplrl 777	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ∈ On)
24		simpr 484	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
25	24	adantl 481	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
26		onelon 6409	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
27	25, 26	sylan 580	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
28		oaword1 8590	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑋))
29	23, 27, 28	syl2anc 584	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑋))
30		oacl 8573	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
31		eloni 6394	. . . . . . . . 9 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
33		eloni 6394	. . . . . . . . 9 ⊢ (𝐶 ∈ On → Ord 𝐶)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
35	32, 34	jca 511	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
36	35	adantl 481	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
37	36	adantr 480	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
38		ordeldif 43271	. . . . 5 ⊢ ((Ord (𝐶 +o 𝐷) ∧ Ord 𝐶) → ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑋))))
39	37, 38	syl 17	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ ((𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷) ∧ 𝐶 ⊆ (𝐶 +o 𝑋))))
40	22, 29, 39	mpbir2and 713	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
41	5, 15, 17, 40	fvun2d 7003	. 2 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})‘(𝐶 +o 𝑋)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}‘(𝐶 +o 𝑋)))
42		eqid 2737	. . . . . 6 ⊢ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)
43		eqid 2737	. . . . . 6 ⊢ (𝐵‘𝑋) = (𝐵‘𝑋)
44		oveq2 7439	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐶 +o 𝑧) = (𝐶 +o 𝑋))
45	44	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)))
46		fveq2 6906	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐵‘𝑧) = (𝐵‘𝑋))
47	46	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐵‘𝑋) = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑋)))
48	45, 47	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑋) ∧ (𝐵‘𝑋) = (𝐵‘𝑋))))
49	48	rspcev 3622	. . . . . 6 ⊢ ((𝑋 ∈ 𝐷 ∧ ((𝐶 +o 𝑋) = (𝐶 +o 𝑋) ∧ (𝐵‘𝑋) = (𝐵‘𝑋))) → ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))
50	42, 43, 49	mpanr12 705	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))
51	50	adantl 481	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))
52		ovex 7464	. . . . . 6 ⊢ (𝐶 +o 𝑋) ∈ V
53		fvex 6919	. . . . . 6 ⊢ (𝐵‘𝑋) ∈ V
54	52, 53	pm3.2i 470	. . . . 5 ⊢ ((𝐶 +o 𝑋) ∈ V ∧ (𝐵‘𝑋) ∈ V)
55		eleq1 2829	. . . . . . 7 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
56		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑧)))
57	56	anbi1d 631	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +o 𝑋) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
58	57	rexbidv 3179	. . . . . . 7 ⊢ (𝑥 = (𝐶 +o 𝑋) → (∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
59	55, 58	anbi12d 632	. . . . . 6 ⊢ (𝑥 = (𝐶 +o 𝑋) → ((𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))))
60		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑦 = (𝐵‘𝑋) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑧)))
61	60	anbi2d 630	. . . . . . . 8 ⊢ (𝑦 = (𝐵‘𝑋) → (((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧))))
62	61	rexbidv 3179	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑋) → (∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧))))
63	62	anbi2d 630	. . . . . 6 ⊢ (𝑦 = (𝐵‘𝑋) → (((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))) ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))))
64	59, 63	opelopabg 5543	. . . . 5 ⊢ (((𝐶 +o 𝑋) ∈ V ∧ (𝐵‘𝑋) ∈ V) → (⟨(𝐶 +o 𝑋), (𝐵‘𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))))
65	54, 64	mp1i 13	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (⟨(𝐶 +o 𝑋), (𝐵‘𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} ↔ ((𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))))
66	40, 51, 65	mpbir2and 713	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ⟨(𝐶 +o 𝑋), (𝐵‘𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))})
67		fnopfvb 6960	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))} Fn ((𝐶 +o 𝐷) ∖ 𝐶) ∧ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵‘𝑋) ↔ ⟨(𝐶 +o 𝑋), (𝐵‘𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
68	15, 40, 67	syl2anc 584	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵‘𝑋) ↔ ⟨(𝐶 +o 𝑋), (𝐵‘𝑋)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))
69	66, 68	mpbird 257	. 2 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))
70	4, 41, 69	3eqtrd 2781	1 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃!weu 2568  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ⟨cop 4632  {copab 5205  dom cdm 5685  Ord word 6383  Oncon0 6384   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433   +_o coa 8503

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-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510

This theorem is referenced by:  tfsconcatfv  43354