Theorem fundcmpsurbijinjpreimafv 47412

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 484	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
2	1	mptexd 7201	. . 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 47388	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)
5	4	adantl 481	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝑃 ∈ V)
6	5	mptexd 7201	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V)
7		ffun 6694	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
8		funimaexg 6606	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
9	7, 8	sylan 580	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V)
10	9	resiexd 7193	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V)
11	2, 6, 10	3jca 1128	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V))
12		ffn 6691	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
13		fveq2 6861	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
14	13	sneqd 4604	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
15	14	imaeq2d 6034	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
16	15	cbvmptv 5214	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))
17	3, 16	fundcmpsurinjlem2 47404	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)
18	12, 17	sylan 580	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)
19		eqid 2730	. . . . . 6 ⊢ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))
20	3, 19	imasetpreimafvbij 47411	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))
21	12, 20	sylan 580	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))
22		f1oi 6841	. . . . . 6 ⊢ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴)
23		f1of1 6802	. . . . . 6 ⊢ (( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴))
24		fimass 6711	. . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
25		f1ss 6764	. . . . . . . 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 593	. . . . . . 7 ⊢ ((( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) ∧ 𝐹:𝐴⟶𝐵) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
27	26	ex 412	. . . . . 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 480	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)
30	18, 21, 29	3jca 1128	. . 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 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴)
32		uniimaprimaeqfv 47387	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
33	31, 32	sylan 580	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
34	33	fveq2d 6865	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))) = (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)))
35	34	mpteq2dva 5203	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))))
36		ffrn 6704	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
37	36	adantr 480	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹:𝐴⟶ran 𝐹)
38	37	funfvima2d 7209	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ (𝐹 “ 𝐴))
39		fvresi 7150	. . . . . . . 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 5203	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
42	35, 41	eqtrd 2765	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
43	12	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐹 Fn 𝐴)
44	1	adantr 480	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑉)
45		simpr 484	. . . . . . 7 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
46	3	preimafvelsetpreimafv 47393	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
47	43, 44, 45, 46	syl3anc 1373	. . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
48		eqidd 2731	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))
49		eqidd 2731	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))))
50		imaeq2 6030	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
51	50	unieqd 4887	. . . . . . 7 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ 𝑦) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
52	51	fveq2d 6865	. . . . . 6 ⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦)) = (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
53	47, 48, 49, 52	fmptco 7104	. . . . 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 6922	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
55	12, 54	sylib 218	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
56	55	adantr 480	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
57	42, 53, 56	3eqtr4rd 2776	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
58		f1of 6803	. . . . . . . . . 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 6651	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
61	60	eqcomd 2736	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
62	61	feq2d 6675	. . . . . . . . 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 47401	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝑃) → ∪ (𝐹 “ 𝑦) ∈ ran 𝐹)
65	63, 64	cofmpt 7107	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))))
66	65	eqcomd 2736	. . . . . 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 5828	. . . 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 2765	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))
70	30, 69	jca 511	. 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 6771	. . . . . 6 ⊢ (𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) → (𝑔:𝐴–onto→𝑃 ↔ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃))
72	71	3ad2ant1 1133	. . . . 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 6791	. . . . . 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 1134	. . . . 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 6754	. . . . . 6 ⊢ (𝑖 = ( I ↾ (𝐹 “ 𝐴)) → (𝑖:(𝐹 “ 𝐴)–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))
76	75	3ad2ant3 1135	. . . . 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 1438	. . . 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 1138	. . . . . . 7 ⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → 𝑖 = ( I ↾ (𝐹 “ 𝐴)))
79		simp2 1137	. . . . . . 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 5831	. . . . . 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 1136	. . . . . 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 5831	. . . . 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 2741	. . . 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 3572	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  {csn 4592  ∪ cuni 4874   ↦ cmpt 5191   I cid 5535  ^◡ccnv 5640  ran crn 5642   ↾ cres 5643   “ cima 5644   ∘ ccom 5645  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  fundcmpsurinjpreimafv  47413  fundcmpsurbijinj  47415