Theorem imasetpreimafvbijlemfo 48103

Description: Lemma for imasetpreimafvbij 48104: 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 48099	. . 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 48086	. . . . . . . . 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 6062	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑝) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
8	7	unieqd 4890	. . . . . . . . . 10 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ 𝑝) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
9	8	eqeq2d 2781	. . . . . . . . 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 48080	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
14	6, 10, 13	rspcedvd 3591	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
15		eqeq1 2774	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
16	15	eqcoms 2778	. . . . . . . 8 ⊢ ((𝐹‘𝑎) = 𝑦 → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
17	16	rexbidv 3196	. . . . . . 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 3173	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
20	8	eqcomd 2776	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = ∪ (𝐹 “ 𝑝))
21	13, 20	sylan9eq 2825	. . . . . . . . . 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 3185	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
24	1	elsetpreimafv 48083	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
25		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
26	25	sneqd 4606	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
27	26	imaeq2d 6066	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
28	27	eqeq2d 2781	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)})))
29	28	cbvrexvw 3251	. . . . . . . . 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 2782	. . . . . . . 8 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
33	32	rexbidv 3196	. . . . . . 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 3173	. . . . 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 2836	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)})
38		fnfun 6639	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
39		fndm 6642	. . . . . . 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 6947	. . . 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 5951	. . . 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 2816	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ran 𝐻 = (𝐹 “ 𝐴))
49		dffo2 6800	. 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 1568   ∈ wcel 2150  {cab 2748  ∃wrex 3096   ⊆ wss 3913  {csn 4594  ∪ cuni 4877   ↦ cmpt 5197  ^◡ccnv 5664  dom cdm 5665  ran crn 5666   “ cima 5668  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  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 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548

This theorem is referenced by:  imasetpreimafvbij  48104