Theorem ofoafg 41245

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 1136	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐷 ∈ On)
2		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐴 ∈ 𝑉)
3		elmapg 8659	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐷 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐷))
4	1, 2, 3	syl2anr 598	. . . 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 1137	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐸 ∈ On)
6		simp2 1137	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐵 ∈ 𝑊)
7		elmapg 8659	. . . . . . . . 9 ⊢ ((𝐸 ∈ On ∧ 𝐵 ∈ 𝑊) → (𝑔 ∈ (𝐸 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐸))
8	5, 6, 7	syl2anr 598	. . . . . . . 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 482	. . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓:𝐴⟶𝐷)
11	10	ffnd 6631	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓 Fn 𝐴)
12	11	adantl 483	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔:𝐵⟶𝐸)
14	13	ffnd 6631	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔 Fn 𝐵)
15	14	adantl 483	. . . . . . . . . . . 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 724	. . . . . . . . . . . 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 724	. . . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
19	12, 15, 16, 17, 18	offn 7578	. . . . . . . . . . 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 1138	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 = (𝐴 ∩ 𝐵))
21	20	fneq2d 6558	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 ↔ (𝑓 ∘f +o 𝑔) Fn (𝐴 ∩ 𝐵)))
22	21	ad2antrr 724	. . . . . . . . . . 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 257	. . . . . . . . . 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 6673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴⟶𝐷 → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
25	24	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
26	25	adantl 483	. . . . . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
28	20, 27	eqsstrdi 3980	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐴)
29		sseqin2 4155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
30	28, 29	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐶) = 𝐶)
31	30	ad2antrr 724	. . . . . . . . . . . . . . . . . 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 6616	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 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 6993	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . 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 6306	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ On ∧ ((𝑓 ↾ 𝐶)‘𝑐) ∈ 𝐷) → ((𝑓 ↾ 𝐶)‘𝑐) ∈ On)
38	36, 34, 37	syl2anc 585	. . . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:𝐵⟶𝐸 → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
40	39	adantl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
41	40	adantl 483	. . . . . . . . . . . . . . . . . 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 4169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
43	20, 42	eqsstrdi 3980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐵)
44		sseqin2 4155	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
45	43, 44	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐵 ∩ 𝐶) = 𝐶)
46	45	ad2antrr 724	. . . . . . . . . . . . . . . . . . 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 6616	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 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 6993	. . . . . . . . . . . . . . . 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 8408	. . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . 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 7314	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ((𝑓 ↾ 𝐶)‘𝑐) → (𝑑 +o 𝐸) = (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸))
54	53	eliuni 4937	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 7584	. . . . . . . . . . . . . . . . . 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 5903	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑓 ↾ 𝐶) = (𝑓 ↾ (𝐴 ∩ 𝐵)))
58	20	reseq2d 5903	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑔 ↾ 𝐶) = (𝑔 ↾ (𝐴 ∩ 𝐵)))
59	57, 58	oveq12d 7325	. . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 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 6806	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 6587	. . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 6587	. . . . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . . . 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 5252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
70	2, 69	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ V)
71	20, 70	eqeltrd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ∈ V)
72	71	ad2antrr 724	. . . . . . . . . . . . . . . . 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 616	. . . . . . . . . . . . . . . 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 7582	. . . . . . . . . . . . . . . 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 683	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))
78	77	ad3antlr 729	. . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . 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 3139	. . . . . . . . . . . 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 7026	. . . . . . . . . . . 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 594	. . . . . . . . . . 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 415	. . . . . . . . . 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 518	. . . . . . . . 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 6306	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
86	85	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
87		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → 𝐸 ∈ On)
88	87	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝐸 ∈ On)
89		oacl 8396	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ On ∧ 𝐸 ∈ On) → (𝑑 +o 𝐸) ∈ On)
90	86, 88, 89	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → (𝑑 +o 𝐸) ∈ On)
91	90	ralrimiva 3139	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
92		iunon 8201	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
93	91, 92	syldan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
94	93	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
95	77, 94	eqeltrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐹 ∈ On)
96	95	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → 𝐹 ∈ On)
97	96	adantr 482	. . . . . . . . . . 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 8660	. . . . . . . . . 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 6462	. . . . . . . . . 10 ⊢ ((𝑓 ∘f +o 𝑔):𝐶⟶𝐹 ↔ ((𝑓 ∘f +o 𝑔) Fn 𝐶 ∧ ran (𝑓 ∘f +o 𝑔) ⊆ 𝐹))
100	98, 99	bitrdi 287	. . . . . . . . 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 257	. . . . . . . 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 458	. . . . . . 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 239	. . . . . 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 3138	. . . . 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 414	. . . 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 239	. . 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 3138	. 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 7859	. . 3 ⊢ ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))) = (𝑓 ∈ (𝐷 ↑m 𝐴), 𝑔 ∈ (𝐸 ↑m 𝐵) ↦ (𝑓 ∘f +o 𝑔))
109	108	fmpo 7940	. 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 217	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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  ∀wral 3061  Vcvv 3437   ∩ cin 3891   ⊆ wss 3892  ∪ ciun 4931   × cxp 5598  ran crn 5601   ↾ cres 5602  Oncon0 6281   Fn wfn 6453  ⟶wf 6454  ‘cfv 6458  (class class class)co 7307   ∘_f cof 7563   +_o coa 8325   ↑_m cmap 8646

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3332  df-rab 3333  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-oadd 8332  df-map 8648

This theorem is referenced by:  ofoaf  41246