Theorem ofoafg 44182

Description: Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)

Assertion

Ref	Expression
ofoafg	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))

Distinct variable groups:   𝐶,𝑑   𝐷,𝑑   𝐸,𝑑

Allowed substitution hints:   𝐴(𝑑)   𝐵(𝑑)   𝐹(𝑑)   𝑉(𝑑)   𝑊(𝑑)

Proof of Theorem ofoafg

Dummy variables 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1154	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐷 ∈ On)
2		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐴 ∈ 𝑉)
3		elmapg 8841	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐷 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐷))
4	1, 2, 3	syl2anr 609	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → (𝑓 ∈ (𝐷 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐷))
5		simp2 1155	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐸 ∈ On)
6		simp2 1155	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐵 ∈ 𝑊)
7		elmapg 8841	. . . . . . . . 9 ⊢ ((𝐸 ∈ On ∧ 𝐵 ∈ 𝑊) → (𝑔 ∈ (𝐸 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐸))
8	5, 6, 7	syl2anr 609	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → (𝑔 ∈ (𝐸 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐸))
9	8	adantr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ 𝑓:𝐴⟶𝐷) → (𝑔 ∈ (𝐸 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐸))
10		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓:𝐴⟶𝐷)
11	10	ffnd 6707	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓 Fn 𝐴)
12	11	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝑓 Fn 𝐴)
13		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔:𝐵⟶𝐸)
14	13	ffnd 6707	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔 Fn 𝐵)
15	14	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝑔 Fn 𝐵)
16	2	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐴 ∈ 𝑉)
17	6	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐵 ∈ 𝑊)
18		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
19	12, 15, 16, 17, 18	offn 7694	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ∘f +o 𝑔) Fn (𝐴 ∩ 𝐵))
20		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 = (𝐴 ∩ 𝐵))
21	20	fneq2d 6630	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 ↔ (𝑓 ∘f +o 𝑔) Fn (𝐴 ∩ 𝐵)))
22	21	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 ↔ (𝑓 ∘f +o 𝑔) Fn (𝐴 ∩ 𝐵)))
23	19, 22	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ∘f +o 𝑔) Fn 𝐶)
24		fresin 6748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴⟶𝐷 → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
25	24	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
26	25	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
27		inss1 4185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
28	20, 27	eqsstrdi 3978	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐴)
29		sseqin2 4172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
30	28, 29	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐶) = 𝐶)
31	30	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝐴 ∩ 𝐶) = 𝐶)
32	31	feq2d 6690	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷 ↔ (𝑓 ↾ 𝐶):𝐶⟶𝐷))
33	26, 32	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ↾ 𝐶):𝐶⟶𝐷)
34	33	ffvelcdmda 7080	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑓 ↾ 𝐶)‘𝑐) ∈ 𝐷)
35	5	ad3antlr 744	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → 𝐸 ∈ On)
36	1	ad3antlr 744	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → 𝐷 ∈ On)
37		onelon 6386	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ On ∧ ((𝑓 ↾ 𝐶)‘𝑐) ∈ 𝐷) → ((𝑓 ↾ 𝐶)‘𝑐) ∈ On)
38	36, 34, 37	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑓 ↾ 𝐶)‘𝑐) ∈ On)
39		fresin 6748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:𝐵⟶𝐸 → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
40	39	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
41	40	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
42		inss2 4186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
43	20, 42	eqsstrdi 3978	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐵)
44		sseqin2 4172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
45	43, 44	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐵 ∩ 𝐶) = 𝐶)
46	45	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝐵 ∩ 𝐶) = 𝐶)
47	46	feq2d 6690	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸 ↔ (𝑔 ↾ 𝐶):𝐶⟶𝐸))
48	41, 47	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑔 ↾ 𝐶):𝐶⟶𝐸)
49	48	ffvelcdmda 7080	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑔 ↾ 𝐶)‘𝑐) ∈ 𝐸)
50		oaordi 8536	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ On ∧ ((𝑓 ↾ 𝐶)‘𝑐) ∈ On) → (((𝑔 ↾ 𝐶)‘𝑐) ∈ 𝐸 → (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸)))
51	50	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 ∈ On ∧ ((𝑓 ↾ 𝐶)‘𝑐) ∈ On) ∧ ((𝑔 ↾ 𝐶)‘𝑐) ∈ 𝐸) → (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸))
52	35, 38, 49, 51	syl21anc 851	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸))
53		oveq1 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ((𝑓 ↾ 𝐶)‘𝑐) → (𝑑 +o 𝐸) = (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸))
54	53	eliuni 4960	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 ↾ 𝐶)‘𝑐) ∈ 𝐷 ∧ (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸)) → (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))
55	34, 52, 54	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)) ∈ ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))
56	12, 15, 16, 17, 18	ofres 7700	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ∘f +o 𝑔) = ((𝑓 ↾ (𝐴 ∩ 𝐵)) ∘f +o (𝑔 ↾ (𝐴 ∩ 𝐵))))
57	20	reseq2d 5976	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑓 ↾ 𝐶) = (𝑓 ↾ (𝐴 ∩ 𝐵)))
58	20	reseq2d 5976	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑔 ↾ 𝐶) = (𝑔 ↾ (𝐴 ∩ 𝐵)))
59	57, 58	oveq12d 7434	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → ((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶)) = ((𝑓 ↾ (𝐴 ∩ 𝐵)) ∘f +o (𝑔 ↾ (𝐴 ∩ 𝐵))))
60	59	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶)) = ((𝑓 ↾ (𝐴 ∩ 𝐵)) ∘f +o (𝑔 ↾ (𝐴 ∩ 𝐵))))
61	56, 60	eqtr4d 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ∘f +o 𝑔) = ((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶)))
62	61	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔)‘𝑐) = (((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶))‘𝑐))
63	62	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑓 ∘f +o 𝑔)‘𝑐) = (((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶))‘𝑐))
64	28	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐶 ⊆ 𝐴)
65	12, 64	fnssresd 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ↾ 𝐶) Fn 𝐶)
66	43	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐶 ⊆ 𝐵)
67	15, 66	fnssresd 6660	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑔 ↾ 𝐶) Fn 𝐶)
68	65, 67	jca 521	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ↾ 𝐶) Fn 𝐶 ∧ (𝑔 ↾ 𝐶) Fn 𝐶))
69		inex1g 5286	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
70	2, 69	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ V)
71	20, 70	eqeltrd 2862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ∈ V)
72	71	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐶 ∈ V)
73	72	anim1i 627	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → (𝐶 ∈ V ∧ 𝑐 ∈ 𝐶))
74		fnfvof 7698	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 ↾ 𝐶) Fn 𝐶 ∧ (𝑔 ↾ 𝐶) Fn 𝐶) ∧ (𝐶 ∈ V ∧ 𝑐 ∈ 𝐶)) → (((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶))‘𝑐) = (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)))
75	68, 73, 74	syl2an2r 698	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → (((𝑓 ↾ 𝐶) ∘f +o (𝑔 ↾ 𝐶))‘𝑐) = (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)))
76	63, 75	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑓 ∘f +o 𝑔)‘𝑐) = (((𝑓 ↾ 𝐶)‘𝑐) +o ((𝑔 ↾ 𝐶)‘𝑐)))
77		simp3 1156	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))
78	77	ad3antlr 744	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))
79	55, 76, 78	3eltr4d 2877	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) ∧ 𝑐 ∈ 𝐶) → ((𝑓 ∘f +o 𝑔)‘𝑐) ∈ 𝐹)
80	79	ralrimiva 3156	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ∀𝑐 ∈ 𝐶 ((𝑓 ∘f +o 𝑔)‘𝑐) ∈ 𝐹)
81		fnfvrnss 7117	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ ∀𝑐 ∈ 𝐶 ((𝑓 ∘f +o 𝑔)‘𝑐) ∈ 𝐹) → ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹)
82	80, 81	sylan2 605	. . . . . . . . . . 11 ⊢ (((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸))) → ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹)
83	82	expcom 419	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 → ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹))
84	23, 83	jcai 526	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹))
85		onelon 6386	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
86	85	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
87		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → 𝐸 ∈ On)
88	87	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝐸 ∈ On)
89		oacl 8525	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ On ∧ 𝐸 ∈ On) → (𝑑 +o 𝐸) ∈ On)
90	86, 88, 89	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → (𝑑 +o 𝐸) ∈ On)
91	90	ralrimiva 3156	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
92		iunon 8331	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
93	91, 92	syldan 603	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
94	93	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
95	77, 94	eqeltrd 2862	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐹 ∈ On)
96	95	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → 𝐹 ∈ On)
97	96	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → 𝐹 ∈ On)
98	97, 72	elmapd 8842	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶) ↔ (𝑓 ∘f +o 𝑔):𝐶⟶𝐹))
99		df-f 6541	. . . . . . . . . 10 ⊢ ((𝑓 ∘f +o 𝑔):𝐶⟶𝐹 ↔ ((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹))
100	98, 99	bitrdi 290	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → ((𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶) ↔ ((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹)))
101	84, 100	mpbird 260	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ (𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸)) → (𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶))
102	101	expr 462	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ 𝑓:𝐴⟶𝐷) → (𝑔:𝐵⟶𝐸 → (𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶)))
103	9, 102	sylbid 243	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ 𝑓:𝐴⟶𝐷) → (𝑔 ∈ (𝐸 ↑m 𝐵) → (𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶)))
104	103	ralrimiv 3155	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) ∧ 𝑓:𝐴⟶𝐷) → ∀𝑔 ∈ (𝐸 ↑m 𝐵)(𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶))
105	104	ex 418	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → (𝑓:𝐴⟶𝐷 → ∀𝑔 ∈ (𝐸 ↑m 𝐵)(𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶)))
106	4, 105	sylbid 243	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → (𝑓 ∈ (𝐷 ↑m 𝐴) → ∀𝑔 ∈ (𝐸 ↑m 𝐵)(𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶)))
107	106	ralrimiv 3155	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ∀𝑓 ∈ (𝐷 ↑m 𝐴)∀𝑔 ∈ (𝐸 ↑m 𝐵)(𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶))
108		ofmres 7984	. . 3 ⊢ ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))) = (𝑓 ∈ (𝐷 ↑m 𝐴), 𝑔 ∈ (𝐸 ↑m 𝐵) ↦ (𝑓 ∘f +o 𝑔))
109	108	fmpo 8068	. 2 ⊢ (∀𝑓 ∈ (𝐷 ↑m 𝐴)∀𝑔 ∈ (𝐸 ↑m 𝐵)(𝑓 ∘f +o 𝑔) ∈ (𝐹 ↑m 𝐶) ↔ ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))
110	107, 109	sylib 221	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∪ ciun 4954   × cxp 5657  ran crn 5660   ↾ cres 5661  Oncon0 6361   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ∘_f cof 7679   +_o coa 8455   ↑_m cmap 8829

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  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-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-oadd 8462  df-map 8831

This theorem is used by:  ofoaf  44183