Theorem ofoafg 43343

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 8812	. . . . 5 ⊢ ((𝐷 ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝐷 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐷))
4	1, 2, 3	syl2anr 597	. . . 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 8812	. . . . . . . . 9 ⊢ ((𝐸 ∈ On ∧ 𝐵 ∈ 𝑊) → (𝑔 ∈ (𝐸 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐸))
8	5, 6, 7	syl2anr 597	. . . . . . . 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 480	. . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓:𝐴⟶𝐷)
11	10	ffnd 6689	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑓 Fn 𝐴)
12	11	adantl 481	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔:𝐵⟶𝐸)
14	13	ffnd 6689	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → 𝑔 Fn 𝐵)
15	14	adantl 481	. . . . . . . . . . . 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 726	. . . . . . . . . . . 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 726	. . . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
19	12, 15, 16, 17, 18	offn 7666	. . . . . . . . . . 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 6612	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → ((𝑓 ∘f +o 𝑔) Fn 𝐶 ↔ (𝑓 ∘f +o 𝑔) Fn (𝐴 ∩ 𝐵)))
22	21	ad2antrr 726	. . . . . . . . . . 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 6729	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴⟶𝐷 → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
25	24	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑓 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶𝐷)
26	25	adantl 481	. . . . . . . . . . . . . . . . 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 4200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
28	20, 27	eqsstrdi 3991	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐴)
29		sseqin2 4186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶)
30	28, 29	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐶) = 𝐶)
31	30	ad2antrr 726	. . . . . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . 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 6357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ On ∧ ((𝑓 ↾ 𝐶)‘𝑐) ∈ 𝐷) → ((𝑓 ↾ 𝐶)‘𝑐) ∈ On)
38	36, 34, 37	syl2anc 584	. . . . . . . . . . . . . . . 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 6729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔:𝐵⟶𝐸 → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
40	39	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴⟶𝐷 ∧ 𝑔:𝐵⟶𝐸) → (𝑔 ↾ 𝐶):(𝐵 ∩ 𝐶)⟶𝐸)
41	40	adantl 481	. . . . . . . . . . . . . . . . . 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 4201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
43	20, 42	eqsstrdi 3991	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ⊆ 𝐵)
44		sseqin2 4186	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
45	43, 44	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝐵 ∩ 𝐶) = 𝐶)
46	45	ad2antrr 726	. . . . . . . . . . . . . . . . . . 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 6672	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . 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 8510	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ((𝑓 ↾ 𝐶)‘𝑐) → (𝑑 +o 𝐸) = (((𝑓 ↾ 𝐶)‘𝑐) +o 𝐸))
54	53	eliuni 4961	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 7672	. . . . . . . . . . . . . . . . . 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 5950	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑓 ↾ 𝐶) = (𝑓 ↾ (𝐴 ∩ 𝐵)))
58	20	reseq2d 5950	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → (𝑔 ↾ 𝐶) = (𝑔 ↾ (𝐴 ∩ 𝐵)))
59	57, 58	oveq12d 7405	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 6642	. . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 6642	. . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . 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 5274	. . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) → 𝐶 ∈ V)
72	71	ad2antrr 726	. . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . 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 7670	. . . . . . . . . . . . . . . 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 685	. . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . 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 3125	. . . . . . . . . . . 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 7093	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 413	. . . . . . . . . 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 516	. . . . . . . . 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 6357	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
86	85	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ On)
87		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → 𝐸 ∈ On)
88	87	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → 𝐸 ∈ On)
89		oacl 8499	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ On ∧ 𝐸 ∈ On) → (𝑑 +o 𝐸) ∈ On)
90	86, 88, 89	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷 ∈ On ∧ 𝐸 ∈ On) ∧ 𝑑 ∈ 𝐷) → (𝑑 +o 𝐸) ∈ On)
91	90	ralrimiva 3125	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On) → ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
92		iunon 8308	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ On ∧ ∀𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On) → ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸) ∈ On)
93	91, 92	syldan 591	. . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸)) → 𝐹 ∈ On)
96	95	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → 𝐹 ∈ On)
97	96	adantr 480	. . . . . . . . . . 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 8813	. . . . . . . . . 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 6515	. . . . . . . . . 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 456	. . . . . . 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 240	. . . . . 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 3124	. . . . 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 412	. . . 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 240	. . 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 3124	. 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 7963	. . 3 ⊢ ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))) = (𝑓 ∈ (𝐷 ↑m 𝐴), 𝑔 ∈ (𝐸 ↑m 𝐵) ↦ (𝑓 ∘f +o 𝑔))
109	108	fmpo 8047	. 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 218	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∪ ciun 4955   × cxp 5636  ran crn 5639   ↾ cres 5640  Oncon0 6332   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   +_o coa 8431   ↑_m cmap 8799

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438  df-map 8801

This theorem is referenced by:  ofoaf  43344