Theorem imasetpreimafvbijlemfo 48042

Description: Lemma for imasetpreimafvbij 48043: 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 48038	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	3	adantr 485	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃⟶(𝐹 “ 𝐴))
5	1	preimafvelsetpreimafv 48025	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
6	5	3expa 1134	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
7		imaeq2 6059	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑝) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
8	7	unieqd 4889	. . . . . . . . . 10 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
9	8	eqeq2d 2780	. . . . . . . . 9 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
10	9	adantl 486	. . . . . . . 8 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
11		uniimaprimaeqfv 48019	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
12	11	adantlr 727	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
13	12	eqcomd 2775	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
14	6, 10, 13	rspcedvd 3592	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
15		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
16	15	eqcoms 2777	. . . . . . . 8 ⊢ ((𝐹‘𝑎) = 𝑦 → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
17	16	rexbidv 3195	. . . . . . 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 3172	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
20	8	eqcomd 2775	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = ∪ (𝐹 “ 𝑝))
21	13, 20	sylan9eq 2824	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
22	21	ex 417	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
23	22	reximdva 3184	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
24	1	elsetpreimafv 48022	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
25		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
26	25	sneqd 4606	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
27	26	imaeq2d 6063	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
28	27	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)})))
29	28	cbvrexvw 3250	. . . . . . . . 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 514	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑝 ∈ 𝑃) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
32		eqeq2 2781	. . . . . . . 8 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
33	32	rexbidv 3195	. . . . . . 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 3172	. . . . 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 2835	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)})
38		fnfun 6636	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
39		fndm 6639	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40		eqimss2 4004	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
41	39, 40	syl 18	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
42	38, 41	jca 520	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
43	42	adantr 485	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
44		dfimafn 6944	. . . 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 5948	. . . 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 2815	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ran 𝐻 = (𝐹 “ 𝐴))
49		dffo2 6797	. 2 ⊢ (𝐻:𝑃–onto→(𝐹 “ 𝐴) ↔ (𝐻:𝑃⟶(𝐹 “ 𝐴) ∧ ran 𝐻 = (𝐹 “ 𝐴)))
50	4, 48, 49	sylanbrc 594	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095   ⊆ wss 3913  {csn 4594  ∪ cuni 4876   ↦ cmpt 5196  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  imasetpreimafvbij  48043