Theorem imasetpreimafvbijlemfo 48305

Description: Lemma for imasetpreimafvbij 48306: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
imasetpreimafvbijlemfo	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑉,𝑝

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥, 𝑧, 𝑝)   𝑉(𝑥, 𝑧)

Proof of Theorem imasetpreimafvbijlemfo

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

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2		fundcmpsurinj.h	. . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
3	1, 2	imasetpreimafvbijlemf 48301	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	3	adantr 486	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃⟶(𝐹 “ 𝐴))
5	1	preimafvelsetpreimafv 48288	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
6	5	3expa 1136	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
7		imaeq2 6052	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑝) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
8	7	unieqd 4880	. . . . . . . . . 10 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
9	8	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
10	9	adantl 487	. . . . . . . 8 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
11		uniimaprimaeqfv 48282	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
12	11	adantlr 728	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
13	12	eqcomd 2766	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
14	6, 10, 13	rspcedvd 3578	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
15		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
16	15	eqcoms 2768	. . . . . . . 8 ⊢ ((𝐹‘𝑎) = 𝑦 → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
17	16	rexbidv 3186	. . . . . . 7 ⊢ ((𝐹‘𝑎) = 𝑦 → (∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝) ↔ ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
18	14, 17	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
19	18	rexlimdva 3163	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
20	8	eqcomd 2766	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = ∪ (𝐹 “ 𝑝))
21	13, 20	sylan9eq 2815	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
22	21	ex 418	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
23	22	reximdva 3175	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
24	1	elsetpreimafv 48285	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
25		fveq2 6878	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
26	25	sneqd 4596	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
27	26	imaeq2d 6056	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
28	27	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)})))
29	28	cbvrexvw 3241	. . . . . . . . 9 ⊢ (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
30	24, 29	sylibr 237	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}))
31	23, 30	impel 515	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑝 ∈ 𝑃) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
32		eqeq2 2772	. . . . . . . 8 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
33	32	rexbidv 3186	. . . . . . 7 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 ↔ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
34	31, 33	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑝 ∈ 𝑃) → (𝑦 = ∪ (𝐹 “ 𝑝) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦))
35	34	rexlimdva 3163	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦))
36	19, 35	impbid 215	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 ↔ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
37	36	abbidv 2826	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)})
38		fnfun 6632	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
39		fndm 6635	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40		eqimss2 3990	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
41	39, 40	syl 18	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
42	38, 41	jca 521	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
43	42	adantr 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
44		dfimafn 6940	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦})
45	43, 44	syl 18	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦})
46	2	rnmpt 5941	. . . 4 ⊢ ran 𝐻 = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)}
47	46	a1i 11	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ran 𝐻 = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)})
48	37, 45, 47	3eqtr4rd 2806	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ran 𝐻 = (𝐹 “ 𝐴))
49		dffo2 6793	. 2 ⊢ (𝐻:𝑃–onto→(𝐹 “ 𝐴) ↔ (𝐻:𝑃⟶(𝐹 “ 𝐴) ∧ ran 𝐻 = (𝐹 “ 𝐴)))
50	4, 48, 49	sylanbrc 595	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533

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-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is used by:  imasetpreimafvbij  48306