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Theorem undifixp 8945

Description: Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.)

**Assertion**
Ref	Expression
undifixp	⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝐺 Allowed substitution hint: 𝐶(𝑥)

**Proof of Theorem undifixp**
Step	Hyp	Ref	Expression
1		unexg 7749	. . 3 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶) → (𝐹 ∪ 𝐺) ∈ V)
2	1	3adant3 1150	. 2 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ V)
3		ixpfn 8914	. . . 4 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → 𝐺 Fn (𝐴 ∖ 𝐵))
4		ixpfn 8914	. . . 4 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → 𝐹 Fn 𝐵)
5		3simpa 1166	. . . . . . . 8 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → (𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵))
6	5	ancomd 467	. . . . . . 7 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → (𝐹 Fn 𝐵 ∧ 𝐺 Fn (𝐴 ∖ 𝐵)))
7		disjdif 4429	. . . . . . 7 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅
8		fnun 6650	. . . . . . 7 ⊢ (((𝐹 Fn 𝐵 ∧ 𝐺 Fn (𝐴 ∖ 𝐵)) ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐵 ∪ (𝐴 ∖ 𝐵)))
9	6, 7, 8	sylancl 598	. . . . . 6 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) Fn (𝐵 ∪ (𝐴 ∖ 𝐵)))
10		undif 4441	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
11	10	biimpi 219	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴)
12	11	eqcomd 2768	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
13	12	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)))
14	13	fneq2d 6630	. . . . . 6 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ∪ 𝐺) Fn 𝐴 ↔ (𝐹 ∪ 𝐺) Fn (𝐵 ∪ (𝐴 ∖ 𝐵))))
15	9, 14	mpbird 260	. . . . 5 ⊢ ((𝐺 Fn (𝐴 ∖ 𝐵) ∧ 𝐹 Fn 𝐵 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) Fn 𝐴)
16	15	3exp 1137	. . . 4 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → (𝐹 Fn 𝐵 → (𝐵 ⊆ 𝐴 → (𝐹 ∪ 𝐺) Fn 𝐴)))
17	3, 4, 16	syl2imc 42	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → (𝐵 ⊆ 𝐴 → (𝐹 ∪ 𝐺) Fn 𝐴)))
18	17	3imp 1128	. 2 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) Fn 𝐴)
19		elixp2 8912	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶))
20	19	simp3bi 1165	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶)
21		fndm 6639	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → dom 𝐺 = (𝐴 ∖ 𝐵))
22		elndif 4083	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ (𝐴 ∖ 𝐵))
23		eleq2 2851	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∖ 𝐵) = dom 𝐺 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝑥 ∈ dom 𝐺))
24	23	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∖ 𝐵) = dom 𝐺 → (¬ 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ¬ 𝑥 ∈ dom 𝐺))
25	24	eqcoms 2770	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐺 = (𝐴 ∖ 𝐵) → (¬ 𝑥 ∈ (𝐴 ∖ 𝐵) ↔ ¬ 𝑥 ∈ dom 𝐺))
26		ndmfv 6914	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑥 ∈ dom 𝐺 → (𝐺‘𝑥) = ∅)
27	25, 26	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (dom 𝐺 = (𝐴 ∖ 𝐵) → (¬ 𝑥 ∈ (𝐴 ∖ 𝐵) → (𝐺‘𝑥) = ∅))
28	21, 22, 27	syl2im 41	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = ∅))
29	28	ralrimiv 3155	. . . . . . . . . . . 12 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → ∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = ∅)
30		uneq2 4112	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑥) = ∅ → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐹‘𝑥) ∪ ∅))
31		un0 4347	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) ∪ ∅) = (𝐹‘𝑥)
32		eqtr 2782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐹‘𝑥) ∪ ∅) ∧ ((𝐹‘𝑥) ∪ ∅) = (𝐹‘𝑥)) → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (𝐹‘𝑥))
33		eleq1 2850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) → ((𝐹‘𝑥) ∈ 𝐶 ↔ ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
34	33	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) → ((𝐹‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
35	34	eqcoms 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (𝐹‘𝑥) → ((𝐹‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
36	32, 35	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐹‘𝑥) ∪ ∅) ∧ ((𝐹‘𝑥) ∪ ∅) = (𝐹‘𝑥)) → ((𝐹‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
37	30, 31, 36	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑥) = ∅ → ((𝐹‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
38	37	com12 33	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ 𝐶 → ((𝐺‘𝑥) = ∅ → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
39	38	ral2imi 3103	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) ∈ 𝐶 → (∀𝑥 ∈ 𝐵 (𝐺‘𝑥) = ∅ → ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
40	20, 29, 39	syl2imc 42	. . . . . . . . . . 11 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
41	3, 40	syl 18	. . . . . . . . . 10 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
42	41	impcom 413	. . . . . . . . 9 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶) → ∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)
43		elixp2 8912	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐴 ∖ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∖ 𝐵)(𝐺‘𝑥) ∈ 𝐶))
44	43	simp3bi 1165	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ (𝐴 ∖ 𝐵)(𝐺‘𝑥) ∈ 𝐶)
45		fndm 6639	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
46		eldifn 4082	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵)
47		eleq2 2851	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = dom 𝐹 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ dom 𝐹))
48	47	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 = dom 𝐹 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ dom 𝐹))
49		ndmfv 6914	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = ∅)
50	48, 49	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = dom 𝐹 → (¬ 𝑥 ∈ 𝐵 → (𝐹‘𝑥) = ∅))
51	50	eqcoms 2770	. . . . . . . . . . . . . 14 ⊢ (dom 𝐹 = 𝐵 → (¬ 𝑥 ∈ 𝐵 → (𝐹‘𝑥) = ∅))
52	45, 46, 51	syl2im 41	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝐹‘𝑥) = ∅))
53	52	ralrimiv 3155	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐵 → ∀𝑥 ∈ (𝐴 ∖ 𝐵)(𝐹‘𝑥) = ∅)
54		uneq1 4111	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑥) = ∅ → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (∅ ∪ (𝐺‘𝑥)))
55		uncom 4108	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅)
56		eqtr 2782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (∅ ∪ (𝐺‘𝑥)) ∧ (∅ ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅)) → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅))
57		un0 4347	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑥) ∪ ∅) = (𝐺‘𝑥)
58		eqtr 2782	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅) ∧ ((𝐺‘𝑥) ∪ ∅) = (𝐺‘𝑥)) → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (𝐺‘𝑥))
59		eleq1 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) → ((𝐺‘𝑥) ∈ 𝐶 ↔ ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
60	59	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) → ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
61	60	eqcoms 2770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (𝐺‘𝑥) → ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
62	58, 61	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅) ∧ ((𝐺‘𝑥) ∪ ∅) = (𝐺‘𝑥)) → ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
63	56, 57, 62	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑥) ∪ (𝐺‘𝑥)) = (∅ ∪ (𝐺‘𝑥)) ∧ (∅ ∪ (𝐺‘𝑥)) = ((𝐺‘𝑥) ∪ ∅)) → ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
64	54, 55, 63	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑥) = ∅ → ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
65	64	com12 33	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑥) ∈ 𝐶 → ((𝐹‘𝑥) = ∅ → ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
66	65	ral2imi 3103	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝐴 ∖ 𝐵)(𝐺‘𝑥) ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∖ 𝐵)(𝐹‘𝑥) = ∅ → ∀𝑥 ∈ (𝐴 ∖ 𝐵)((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
67	44, 53, 66	syl2imc 42	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐵 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ (𝐴 ∖ 𝐵)((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
68	4, 67	syl 18	. . . . . . . . . 10 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ (𝐴 ∖ 𝐵)((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
69	68	imp 412	. . . . . . . . 9 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶) → ∀𝑥 ∈ (𝐴 ∖ 𝐵)((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)
70		ralunb 4146	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶 ↔ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶 ∧ ∀𝑥 ∈ (𝐴 ∖ 𝐵)((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
71	42, 69, 70	sylanbrc 595	. . . . . . . 8 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶) → ∀𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)
72	71	ex 418	. . . . . . 7 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
73		raleq 3318	. . . . . . . 8 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶 ↔ ∀𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
74	73	imbi2d 343	. . . . . . 7 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → ((𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶) ↔ (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ (𝐵 ∪ (𝐴 ∖ 𝐵))((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)))
75	72, 74	imbitrrid 249	. . . . . 6 ⊢ (𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)))
76	75	eqcoms 2770	. . . . 5 ⊢ ((𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴 → (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)))
77	10, 76	sylbi 220	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)))
78	77	3imp231 1130	. . 3 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶)
79		df-fn 6540	. . . . . 6 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)))
80		df-fn 6540	. . . . . . . 8 ⊢ (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
81		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐵) → Fun 𝐹)
82		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → Fun 𝐺)
83	81, 82	anim12i 625	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵))) → (Fun 𝐹 ∧ Fun 𝐺))
84	83	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) ∧ 𝐵 ⊆ 𝐴) → (Fun 𝐹 ∧ Fun 𝐺))
85		ineq12 4164	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝐹 = 𝐵 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → (dom 𝐹 ∩ dom 𝐺) = (𝐵 ∩ (𝐴 ∖ 𝐵)))
86	85, 7	eqtrdi 2813	. . . . . . . . . . . . . 14 ⊢ ((dom 𝐹 = 𝐵 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
87	86	ad2ant2l 759	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵))) → (dom 𝐹 ∩ dom 𝐺) = ∅)
88	87	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅)
89		fvun 6972	. . . . . . . . . . . 12 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)))
90	84, 88, 89	syl2anc 596	. . . . . . . . . . 11 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ∪ 𝐺)‘𝑥) = ((𝐹‘𝑥) ∪ (𝐺‘𝑥)))
91	90	eleq1d 2847	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) ∧ 𝐵 ⊆ 𝐴) → (((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
92	91	ralbidv 3187	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐵) ∧ (Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
93	92	3exp 1137	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = 𝐵) → ((Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))))
94	80, 93	sylbi 220	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 → ((Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))))
95	94	com12 33	. . . . . 6 ⊢ ((Fun 𝐺 ∧ dom 𝐺 = (𝐴 ∖ 𝐵)) → (𝐹 Fn 𝐵 → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))))
96	79, 95	sylbi 220	. . . . 5 ⊢ (𝐺 Fn (𝐴 ∖ 𝐵) → (𝐹 Fn 𝐵 → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))))
97	3, 4, 96	syl2imc 42	. . . 4 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 → (𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))))
98	97	3imp 1128	. . 3 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) ∪ (𝐺‘𝑥)) ∈ 𝐶))
99	78, 98	mpbird 260	. 2 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶)
100		elixp2 8912	. 2 ⊢ ((𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ((𝐹 ∪ 𝐺) ∈ V ∧ (𝐹 ∪ 𝐺) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((𝐹 ∪ 𝐺)‘𝑥) ∈ 𝐶))
101	2, 18, 99, 100	syl3anbrc 1362	1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 dom cdm 5659 Fun wfun 6531 Fn wfn 6532 ‘cfv 6537 Xcixp 8908

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-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 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-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-id 5554 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-iota 6493 df-fun 6539 df-fn 6540 df-fv 6545 df-ixp 8909

This theorem is used by: ptuncnv 24039 ptunhmeo 24040

W3C validator