Theorem imasetpreimafvbijlemfo 47379

Description: Lemma for imasetpreimafvbij 47380: 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 47375	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
4	3	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃⟶(𝐹 “ 𝐴))
5	1	preimafvelsetpreimafv 47362	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
6	5	3expa 1118	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃)
7		imaeq2 6016	. . . . . . . . . . 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 2740	. . . . . . . . 9 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
10	9	adantl 481	. . . . . . . 8 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → ((𝐹‘𝑎) = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))
11		uniimaprimaeqfv 47356	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
12	11	adantlr 715	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎))
13	12	eqcomd 2735	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) = ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))
14	6, 10, 13	rspcedvd 3587	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
15		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
16	15	eqcoms 2737	. . . . . . . 8 ⊢ ((𝐹‘𝑎) = 𝑦 → (𝑦 = ∪ (𝐹 “ 𝑝) ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
17	16	rexbidv 3157	. . . . . . 7 ⊢ ((𝐹‘𝑎) = 𝑦 → (∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝) ↔ ∃𝑝 ∈ 𝑃 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
18	14, 17	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
19	18	rexlimdva 3134	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 → ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
20	8	eqcomd 2735	. . . . . . . . . . 11 ⊢ (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = ∪ (𝐹 “ 𝑝))
21	13, 20	sylan9eq 2784	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) ∧ 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)})) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
22	21	ex 412	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
23	22	reximdva 3146	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
24	1	elsetpreimafv 47359	. . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
25		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
26	25	sneqd 4597	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)})
27	26	imaeq2d 6020	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
28	27	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)})))
29	28	cbvrexvw 3214	. . . . . . . . 9 ⊢ (∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}) ↔ ∃𝑥 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑥)}))
30	24, 29	sylibr 234	. . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ∃𝑎 ∈ 𝐴 𝑝 = (◡𝐹 “ {(𝐹‘𝑎)}))
31	23, 30	impel 505	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑝 ∈ 𝑃) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝))
32		eqeq2 2741	. . . . . . . 8 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
33	32	rexbidv 3157	. . . . . . 7 ⊢ (𝑦 = ∪ (𝐹 “ 𝑝) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 ↔ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = ∪ (𝐹 “ 𝑝)))
34	31, 33	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) ∧ 𝑝 ∈ 𝑃) → (𝑦 = ∪ (𝐹 “ 𝑝) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦))
35	34	rexlimdva 3134	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝) → ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦))
36	19, 35	impbid 212	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦 ↔ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)))
37	36	abbidv 2795	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝 ∈ 𝑃 𝑦 = ∪ (𝐹 “ 𝑝)})
38		fnfun 6600	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
39		fndm 6603	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40		eqimss2 4003	. . . . . . 7 ⊢ (dom 𝐹 = 𝐴 → 𝐴 ⊆ dom 𝐹)
41	39, 40	syl 17	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐴 ⊆ dom 𝐹)
42	38, 41	jca 511	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
43	42	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹))
44		dfimafn 6905	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦})
45	43, 44	syl 17	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑎 ∈ 𝐴 (𝐹‘𝑎) = 𝑦})
46	2	rnmpt 5910	. . . 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 2775	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → ran 𝐻 = (𝐹 “ 𝐴))
49		dffo2 6758	. 2 ⊢ (𝐻:𝑃–onto→(𝐹 “ 𝐴) ↔ (𝐻:𝑃⟶(𝐹 “ 𝐴) ∧ ran 𝐻 = (𝐹 “ 𝐴)))
50	4, 48, 49	sylanbrc 583	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3911  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  –onto→wfo 6497  ‘cfv 6499

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by:  imasetpreimafvbij  47380