Theorem tfsconcatfv2 44532

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 44529	. . . 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 6907	. . 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 486	. 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 787	. . 3 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐴 Fn 𝐶)
6		simplrl 789	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐶 ∈ On)
7		simplrr 790	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝐷 ∈ On)
8		simpr 490	. . . . . . 7 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶)) → 𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶))
9		tfsconcatlem 44528	. . . . . . 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 1399	. . . . . 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 3158	. . . . 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 2763	. . . . . 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 6695	. . . . 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 486	. . 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 4429	. . . 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		pm3.22 465	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
19	18	adantl 487	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐷 ∈ On ∧ 𝐶 ∈ On))
20		oaordi 8579	. . . . . 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 412	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (𝐶 +o 𝑋) ∈ (𝐶 +o 𝐷))
23		simplrl 789	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ∈ On)
24		simpr 490	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → 𝐷 ∈ On)
25	24	adantl 487	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → 𝐷 ∈ On)
26		onelon 6402	. . . . . 6 ⊢ ((𝐷 ∈ On ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
27	25, 26	sylan 592	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝑋 ∈ On)
28		oaword1 8585	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝑋 ∈ On) → 𝐶 ⊆ (𝐶 +o 𝑋))
29	23, 27, 28	syl2anc 596	. . . 4 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → 𝐶 ⊆ (𝐶 +o 𝑋))
30		oacl 8568	. . . . . . . . 9 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (𝐶 +o 𝐷) ∈ On)
31		eloni 6387	. . . . . . . . 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 6387	. . . . . . . . 9 ⊢ (𝐶 ∈ On → Ord 𝐶)
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → Ord 𝐶)
35	32, 34	jca 521	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
36	35	adantl 487	. . . . . 6 ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
37	36	adantr 486	. . . . 5 ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → (Ord (𝐶 +o 𝐷) ∧ Ord 𝐶))
38		ordeldif 44450	. . . . 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 726	. . 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 6999	. 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 2763	. . . . . 6 ⊢ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)
43		eqid 2763	. . . . . 6 ⊢ (𝐵‘𝑋) = (𝐵‘𝑋)
44		oveq2 7448	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐶 +o 𝑧) = (𝐶 +o 𝑋))
45	44	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑋)))
46		fveq2 6905	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝐵‘𝑧) = (𝐵‘𝑋))
47	46	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝐵‘𝑋) = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑋)))
48	45, 47	anbi12d 644	. . . . . . 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 718	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → ∃𝑧 ∈ 𝐷 ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ (𝐵‘𝑋) = (𝐵‘𝑧)))
51	50	adantl 487	. . . 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 7473	. . . . . 6 ⊢ (𝐶 +o 𝑋) ∈ V
53		fvex 6918	. . . . . 6 ⊢ (𝐵‘𝑋) ∈ V
54	52, 53	pm3.2i 476	. . . . 5 ⊢ ((𝐶 +o 𝑋) ∈ V ∧ (𝐵‘𝑋) ∈ V)
55		eleq1 2852	. . . . . . 7 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 ∈ ((𝐶 +o 𝐷) ∖ 𝐶) ↔ (𝐶 +o 𝑋) ∈ ((𝐶 +o 𝐷) ∖ 𝐶)))
56		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑥 = (𝐶 +o 𝑋) → (𝑥 = (𝐶 +o 𝑧) ↔ (𝐶 +o 𝑋) = (𝐶 +o 𝑧)))
57	56	anbi1d 643	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +o 𝑋) → ((𝑥 = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)) ↔ ((𝐶 +o 𝑋) = (𝐶 +o 𝑧) ∧ 𝑦 = (𝐵‘𝑧))))
58	57	rexbidv 3190	. . . . . . 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 644	. . . . . 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 2767	. . . . . . . . 9 ⊢ (𝑦 = (𝐵‘𝑋) → (𝑦 = (𝐵‘𝑧) ↔ (𝐵‘𝑋) = (𝐵‘𝑧)))
61	60	anbi2d 642	. . . . . . . 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 3190	. . . . . . 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 642	. . . . . 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 5517	. . . . 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 726	. . 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 6956	. . . 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 596	. . 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 2802	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 401   = wceq 1570   ∈ wcel 2147  ∃!weu 2596  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ⟨cop 4838  {copab 5170  dom cdm 5655  Ord word 6376  Oncon0 6377   Fn wfn 6550  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442   +_o coa 8498

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-oadd 8505

This theorem is used by:  tfsconcatfv  44533