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Theorem imasetpreimafvbijlemfo 46073
Description: Lemma for imasetpreimafvbij 46074: 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 46069 . . 3 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
43adantr 482 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃⟶(𝐹𝐴))
51preimafvelsetpreimafv 46056 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴𝑉𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
653expa 1119 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
7 imaeq2 6056 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
87unieqd 4923 . . . . . . . . . 10 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
98eqeq2d 2744 . . . . . . . . 9 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
109adantl 483 . . . . . . . 8 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
11 uniimaprimaeqfv 46050 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1211adantlr 714 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1312eqcomd 2739 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
146, 10, 13rspcedvd 3615 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝))
15 eqeq1 2737 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1615eqcoms 2741 . . . . . . . 8 ((𝐹𝑎) = 𝑦 → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1716rexbidv 3179 . . . . . . 7 ((𝐹𝑎) = 𝑦 → (∃𝑝𝑃 𝑦 = (𝐹𝑝) ↔ ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝)))
1814, 17syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ((𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
1918rexlimdva 3156 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
208eqcomd 2739 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑝))
2113, 20sylan9eq 2793 . . . . . . . . . 10 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → (𝐹𝑎) = (𝐹𝑝))
2221ex 414 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑎) = (𝐹𝑝)))
2322reximdva 3169 . . . . . . . 8 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
241elsetpreimafv 46053 . . . . . . . . 9 (𝑝𝑃 → ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
25 fveq2 6892 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
2625sneqd 4641 . . . . . . . . . . . 12 (𝑎 = 𝑥 → {(𝐹𝑎)} = {(𝐹𝑥)})
2726imaeq2d 6060 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐹 “ {(𝐹𝑎)}) = (𝐹 “ {(𝐹𝑥)}))
2827eqeq2d 2744 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ 𝑝 = (𝐹 “ {(𝐹𝑥)})))
2928cbvrexvw 3236 . . . . . . . . 9 (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
3024, 29sylibr 233 . . . . . . . 8 (𝑝𝑃 → ∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}))
3123, 30impel 507 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝))
32 eqeq2 2745 . . . . . . . 8 (𝑦 = (𝐹𝑝) → ((𝐹𝑎) = 𝑦 ↔ (𝐹𝑎) = (𝐹𝑝)))
3332rexbidv 3179 . . . . . . 7 (𝑦 = (𝐹𝑝) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
3431, 33syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → (𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3534rexlimdva 3156 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑝𝑃 𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3619, 35impbid 211 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
3736abbidv 2802 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
38 fnfun 6650 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
39 fndm 6653 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40 eqimss2 4042 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
4139, 40syl 17 . . . . . 6 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
4238, 41jca 513 . . . . 5 (𝐹 Fn 𝐴 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
4342adantr 482 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
44 dfimafn 6955 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
4543, 44syl 17 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
462rnmpt 5955 . . . 4 ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)}
4746a1i 11 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
4837, 45, 473eqtr4rd 2784 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = (𝐹𝐴))
49 dffo2 6810 . 2 (𝐻:𝑃onto→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ran 𝐻 = (𝐹𝐴)))
504, 48, 49sylanbrc 584 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  wss 3949  {csn 4629   cuni 4909  cmpt 5232  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680  Fun wfun 6538   Fn wfn 6539  wf 6540  ontowfo 6542  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552
This theorem is referenced by:  imasetpreimafvbij  46074
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