Theorem fundcmpsurbijinjpreimafv 46009

Description: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)

Hypothesis

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

Ref	Expression
fundcmpsurbijinjpreimafv	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝐴,𝑖,𝑔,ℎ   𝐵,𝑔,ℎ,𝑖   𝑥,𝐵,𝑧   𝑖,𝐹,𝑔,ℎ   𝑃,𝑖,𝑔,ℎ   𝑥,𝑃,𝑔,ℎ   𝑥,𝑉

Allowed substitution hints:   𝑃(𝑧)   𝑉(𝑧,𝑔,ℎ,𝑖)

Proof of Theorem fundcmpsurbijinjpreimafv

Dummy variables 𝑎 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 486	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
2	1	mptexd 7220	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V)
3		fundcmpsurinj.p	. . . . . 6 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
4	3	setpreimafvex 45985	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)
5	4	adantl 483	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝑃 ∈ V)
6	5	mptexd 7220	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V)
7		ffun 6716	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
8		funimaexg 6630	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
9	7, 8	sylan 581	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
10	9	resiexd 7212	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V)
11	2, 6, 10	3jca 1129	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V))
12		ffn 6713	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
13		fveq2 6887	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
14	13	sneqd 4638	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
15	14	imaeq2d 6056	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
16	15	cbvmptv 5259	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))
17	3, 16	fundcmpsurinjlem2 46001	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)
18	12, 17	sylan 581	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)
19		eqid 2733	. . . . . 6 ⊢ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))
20	3, 19	imasetpreimafvbij 46008	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))
21	12, 20	sylan 581	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))
22		f1oi 6867	. . . . . 6 ⊢ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴)
23		f1of1 6828	. . . . . 6 ⊢ (( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴))
24		fimass 6734	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
25		f1ss 6789	. . . . . . . 8 ⊢ ((( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
26	24, 25	sylan2 594	. . . . . . 7 ⊢ ((( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) ∧ 𝐹:𝐴⟶𝐵) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
27	26	ex 414	. . . . . 6 ⊢ (( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) → (𝐹:𝐴⟶𝐵 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
28	22, 23, 27	mp2b 10	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
29	28	adantr 482	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
30	18, 21, 29	3jca 1129	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
31	12	adantr 482	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴)
32		uniimaprimaeqfv 45984	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
33	31, 32	sylan 581	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
34	33	fveq2d 6891	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))) = (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)))
35	34	mpteq2dva 5246	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))))
36		ffrn 6727	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
37	36	adantr 482	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹:𝐴⟶ran 𝐹)
38	37	funfvima2d 7228	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ (𝐹 “ 𝐴))
39		fvresi 7165	. . . . . . . 8 ⊢ ((𝐹‘𝑎) ∈ (𝐹 “ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)) = (𝐹‘𝑎))
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)) = (𝐹‘𝑎))
41	40	mpteq2dva 5246	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
42	35, 41	eqtrd 2773	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
43	12	ad2antrr 725	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐹 Fn 𝐴)
44	1	adantr 482	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑉)
45		simpr 486	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
46	3	preimafvelsetpreimafv 45990	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
47	43, 44, 45, 46	syl3anc 1372	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
48		eqidd 2734	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))
49		eqidd 2734	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))))
50		imaeq2 6052	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
51	50	unieqd 4920	. . . . . . 7 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ 𝑦) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
52	51	fveq2d 6891	. . . . . 6 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦)) = (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
53	47, 48, 49, 52	fmptco 7121	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) = (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))))
54		dffn5 6946	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
55	12, 54	sylib 217	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
56	55	adantr 482	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
57	42, 53, 56	3eqtr4rd 2784	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
58		f1of 6829	. . . . . . . . . 10 ⊢ (( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴))
59	22, 58	mp1i 13	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴))
60		fnima 6676	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
61	60	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
62	61	feq2d 6699	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (( I ↾ (𝐹 “ 𝐴)):ran 𝐹⟶(𝐹 “ 𝐴) ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴)))
63	59, 62	mpbird 257	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → ( I ↾ (𝐹 “ 𝐴)):ran 𝐹⟶(𝐹 “ 𝐴))
64	3	uniimaelsetpreimafv 45998	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝑃) → ∪ (𝐹 “ 𝑦) ∈ ran 𝐹)
65	63, 64	cofmpt 7124	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))))
66	65	eqcomd 2739	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))))
67	31, 66	syl 17	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))))
68	67	coeq1d 5858	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
69	57, 68	eqtrd 2773	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
70	30, 69	jca 513	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))))
71		foeq1 6797	. . . . . 6 ⊢ (𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) → (𝑔:𝐴–onto→𝑃 ↔ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃))
72	71	3ad2ant1 1134	. . . . 5 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑔:𝐴–onto→𝑃 ↔ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃))
73		f1oeq1 6817	. . . . . 6 ⊢ (ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) → (ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ↔ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴)))
74	73	3ad2ant2 1135	. . . . 5 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ↔ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴)))
75		f1eq1 6778	. . . . . 6 ⊢ (𝑖 = ( I ↾ (𝐹 “ 𝐴)) → (𝑖:(𝐹 “ 𝐴)–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
76	75	3ad2ant3 1136	. . . . 5 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑖:(𝐹 “ 𝐴)–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
77	72, 74, 76	3anbi123d 1437	. . . 4 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ↔ ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)))
78		simp3 1139	. . . . . . 7 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → 𝑖 = ( I ↾ (𝐹 “ 𝐴)))
79		simp2 1138	. . . . . . 7 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)))
80	78, 79	coeq12d 5861	. . . . . 6 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑖 ∘ ℎ) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))))
81		simp1 1137	. . . . . 6 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → 𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))
82	80, 81	coeq12d 5861	. . . . 5 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ((𝑖 ∘ ℎ) ∘ 𝑔) = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
83	82	eqeq2d 2744	. . . 4 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔) ↔ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))))
84	77, 83	anbi12d 632	. . 3 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))))
85	84	spc3egv 3592	. 2 ⊢ (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V) → ((((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))))
86	11, 70, 85	sylc 65	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∃wrex 3071  Vcvv 3475   ⊆ wss 3946  {csn 4626  ∪ cuni 4906   ↦ cmpt 5229   I cid 5571  ^◡ccnv 5673  ran crn 5675   ↾ cres 5676   “ cima 5677   ∘ ccom 5678  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535  –1-1→wf1 6536  –onto→wfo 6537  –1-1-onto→wf1o 6538  ‘cfv 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547

This theorem is referenced by:  fundcmpsurinjpreimafv  46010  fundcmpsurbijinj  46012