Theorem tfsconcatfv2 44095

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 44092	. . . 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 6883	. . 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 485	. 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 786	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐴 Fn 𝐶)
6		simplrl 788	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
7		simplrr 789	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
8		simpr 489	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
9		tfsconcatlem 44091	. . . . . . 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 1397	. . . . . 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 3156	. . . . 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 2762	. . . . . 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 6672	. . . . 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 221	. . . 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 485	. . 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 4432	. . . 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		pm3.22 464	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
19	18	adantl 486	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
20		oaordi 8529	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ On) → (𝑋 ∈ 𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
21	19, 20	syl 18	. . . . 5 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝑋 ∈ 𝐷 → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷)))
22	21	imp 411	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷))
23		simplrl 788	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ∈ On)
24		simpr 489	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
25	24	adantl 486	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
26		onelon 6385	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
27	25, 26	sylan 591	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
28		oaword1 8535	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑋))
29	23, 27, 28	syl2anc 595	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑋))
30		oacl 8518	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
31		eloni 6370	. . . . . . . . 9 ⊢ ((𝐶 +o 𝐷) ∈ On → Ord (𝐶 +o 𝐷))
32	30, 31	syl 18	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord (𝐶 +o 𝐷))
33		eloni 6370	. . . . . . . . 9 ⊢ (𝐶 ∈ On → Ord 𝐶)
34	33	adantr 485	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
35	32, 34	jca 520	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
36	35	adantl 486	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
37	36	adantr 485	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
38		ordeldif 44013	. . . . 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 18	. . . 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 725	. . 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 6975	. 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 2762	. . . . . 6 ⊢ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)
43		eqid 2762	. . . . . 6 ⊢ (𝐵‘𝑋) = (𝐵‘𝑋)
44		oveq2 7420	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐶 +o 𝑧) = (𝐶 +o 𝑋))
45	44	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)))
46		fveq2 6881	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐵‘𝑧) = (𝐵‘𝑋))
47	46	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐵‘𝑋) = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑋)))
48	45, 47	anbi12d 643	. . . . . . 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 3580	. . . . . 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 717	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))
51	50	adantl 486	. . . 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 7445	. . . . . 6 ⊢ (𝐶 +o 𝑋) ∈ V
53		fvex 6894	. . . . . 6 ⊢ (𝐵‘𝑋) ∈ V
54	52, 53	pm3.2i 475	. . . . 5 ⊢ ((𝐶 +o 𝑋) ∈ V ∧ (𝐵‘𝑋) ∈ V)
55		eleq1 2850	. . . . . . 7 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
56		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑧)))
57	56	anbi1d 642	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +o 𝑋) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
58	57	rexbidv 3188	. . . . . . 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 643	. . . . . 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 2766	. . . . . . . . 9 ⊢ (𝑦 = (𝐵‘𝑋) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑧)))
61	60	anbi2d 641	. . . . . . . 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 3188	. . . . . . 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 641	. . . . . 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 5522	. . . . 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 14	. . . 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 725	. . 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 6932	. . . 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 595	. . 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 260	. 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 2801	1 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +o 𝑋)) = (𝐵‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃!weu 2595  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ⟨cop 4594  {copab 5172  dom cdm 5660  Ord word 6359  Oncon0 6360   Fn wfn 6531  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414   +_o coa 8448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-oadd 8455

This theorem is used by:  tfsconcatfv  44096