Theorem tfsconcatfv2 43438

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 43435	. . . 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 6830	. . 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 774	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐴 Fn 𝐶)
6		simplrl 776	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
7		simplrr 777	. . . . . . 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 43434	. . . . . . 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 3124	. . . . 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 2731	. . . . . 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 6624	. . . . 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 4421	. . . 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 8467	. . . . . 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 776	. . . . 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 6337	. . . . . 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 8473	. . . . 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 8456	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
31		eloni 6322	. . . . . . . . 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 6322	. . . . . . . . 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 43356	. . . . 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 6922	. 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 2731	. . . . . 6 ⊢ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)
43		eqid 2731	. . . . . 6 ⊢ (𝐵‘𝑋) = (𝐵‘𝑋)
44		oveq2 7360	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐶 +o 𝑧) = (𝐶 +o 𝑋))
45	44	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)))
46		fveq2 6828	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐵‘𝑧) = (𝐵‘𝑋))
47	46	eqeq2d 2742	. . . . . . . 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 3572	. . . . . 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 7385	. . . . . 6 ⊢ (𝐶 +o 𝑋) ∈ V
53		fvex 6841	. . . . . 6 ⊢ (𝐵‘𝑋) ∈ V
54	52, 53	pm3.2i 470	. . . . 5 ⊢ ((𝐶 +o 𝑋) ∈ V ∧ (𝐵‘𝑋) ∈ V)
55		eleq1 2819	. . . . . . 7 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
56		eqeq1 2735	. . . . . . . . 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 3156	. . . . . . 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 2735	. . . . . . . . 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 3156	. . . . . . 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 5481	. . . . 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 6879	. . . 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 2770	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 1541   ∈ wcel 2111  ∃!weu 2563  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4282  ⟨cop 4581  {copab 5155  dom cdm 5619  Ord word 6311  Oncon0 6312   Fn wfn 6482  ‘cfv 6487  (class class class)co 7352   ∈ cmpo 7354   +_o coa 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-oadd 8395

This theorem is referenced by:  tfsconcatfv  43439