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Theorem imasetpreimafvbijlemfo 44290
 Description: Lemma for imasetpreimafvbij 44291: 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 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.h 𝐻 = (𝑝𝑃 (𝐹𝑝))
Assertion
Ref Expression
imasetpreimafvbijlemfo ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
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.
StepHypRef Expression
1 fundcmpsurinj.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
2 fundcmpsurinj.h . . . 4 𝐻 = (𝑝𝑃 (𝐹𝑝))
31, 2imasetpreimafvbijlemf 44286 . . 3 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
43adantr 484 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃⟶(𝐹𝐴))
51preimafvelsetpreimafv 44273 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴𝑉𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
653expa 1115 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
7 imaeq2 5897 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
87unieqd 4812 . . . . . . . . . 10 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
98eqeq2d 2769 . . . . . . . . 9 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
109adantl 485 . . . . . . . 8 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
11 uniimaprimaeqfv 44267 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1211adantlr 714 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1312eqcomd 2764 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
146, 10, 13rspcedvd 3544 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝))
15 eqeq1 2762 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1615eqcoms 2766 . . . . . . . 8 ((𝐹𝑎) = 𝑦 → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1716rexbidv 3221 . . . . . . 7 ((𝐹𝑎) = 𝑦 → (∃𝑝𝑃 𝑦 = (𝐹𝑝) ↔ ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝)))
1814, 17syl5ibrcom 250 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ((𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
1918rexlimdva 3208 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
208eqcomd 2764 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑝))
2113, 20sylan9eq 2813 . . . . . . . . . 10 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → (𝐹𝑎) = (𝐹𝑝))
2221ex 416 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑎) = (𝐹𝑝)))
2322reximdva 3198 . . . . . . . 8 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
241elsetpreimafv 44270 . . . . . . . . 9 (𝑝𝑃 → ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
25 fveq2 6658 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
2625sneqd 4534 . . . . . . . . . . . 12 (𝑎 = 𝑥 → {(𝐹𝑎)} = {(𝐹𝑥)})
2726imaeq2d 5901 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐹 “ {(𝐹𝑎)}) = (𝐹 “ {(𝐹𝑥)}))
2827eqeq2d 2769 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ 𝑝 = (𝐹 “ {(𝐹𝑥)})))
2928cbvrexvw 3362 . . . . . . . . 9 (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
3024, 29sylibr 237 . . . . . . . 8 (𝑝𝑃 → ∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}))
3123, 30impel 509 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝))
32 eqeq2 2770 . . . . . . . 8 (𝑦 = (𝐹𝑝) → ((𝐹𝑎) = 𝑦 ↔ (𝐹𝑎) = (𝐹𝑝)))
3332rexbidv 3221 . . . . . . 7 (𝑦 = (𝐹𝑝) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
3431, 33syl5ibrcom 250 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → (𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3534rexlimdva 3208 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑝𝑃 𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3619, 35impbid 215 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
3736abbidv 2822 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
38 fnfun 6434 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
39 fndm 6436 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40 eqimss2 3949 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
4139, 40syl 17 . . . . . 6 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
4238, 41jca 515 . . . . 5 (𝐹 Fn 𝐴 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
4342adantr 484 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
44 dfimafn 6716 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
4543, 44syl 17 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
462rnmpt 5796 . . . 4 ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)}
4746a1i 11 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
4837, 45, 473eqtr4rd 2804 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = (𝐹𝐴))
49 dffo2 6580 . 2 (𝐻:𝑃onto→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ran 𝐻 = (𝐹𝐴)))
504, 48, 49sylanbrc 586 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ⊆ wss 3858  {csn 4522  ∪ cuni 4798   ↦ cmpt 5112  ◡ccnv 5523  dom cdm 5524  ran crn 5525   “ cima 5527  Fun wfun 6329   Fn wfn 6330  ⟶wf 6331  –onto→wfo 6333  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343 This theorem is referenced by:  imasetpreimafvbij  44291
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