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Theorem imasetpreimafvbijlemfo 46373
Description: Lemma for imasetpreimafvbij 46374: 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 46369 . . 3 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
43adantr 479 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃⟶(𝐹𝐴))
51preimafvelsetpreimafv 46356 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴𝑉𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
653expa 1116 . . . . . . . 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 2741 . . . . . . . . 9 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
109adantl 480 . . . . . . . 8 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
11 uniimaprimaeqfv 46350 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1211adantlr 711 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1312eqcomd 2736 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
146, 10, 13rspcedvd 3615 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝))
15 eqeq1 2734 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1615eqcoms 2738 . . . . . . . 8 ((𝐹𝑎) = 𝑦 → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1716rexbidv 3176 . . . . . . 7 ((𝐹𝑎) = 𝑦 → (∃𝑝𝑃 𝑦 = (𝐹𝑝) ↔ ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝)))
1814, 17syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ((𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
1918rexlimdva 3153 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
208eqcomd 2736 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑝))
2113, 20sylan9eq 2790 . . . . . . . . . 10 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → (𝐹𝑎) = (𝐹𝑝))
2221ex 411 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑎) = (𝐹𝑝)))
2322reximdva 3166 . . . . . . . 8 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
241elsetpreimafv 46353 . . . . . . . . 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 2741 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ 𝑝 = (𝐹 “ {(𝐹𝑥)})))
2928cbvrexvw 3233 . . . . . . . . 9 (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
3024, 29sylibr 233 . . . . . . . 8 (𝑝𝑃 → ∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}))
3123, 30impel 504 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝))
32 eqeq2 2742 . . . . . . . 8 (𝑦 = (𝐹𝑝) → ((𝐹𝑎) = 𝑦 ↔ (𝐹𝑎) = (𝐹𝑝)))
3332rexbidv 3176 . . . . . . 7 (𝑦 = (𝐹𝑝) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
3431, 33syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → (𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3534rexlimdva 3153 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑝𝑃 𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3619, 35impbid 211 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
3736abbidv 2799 . . 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 510 . . . . 5 (𝐹 Fn 𝐴 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
4342adantr 479 . . . 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 2781 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = (𝐹𝐴))
49 dffo2 6810 . 2 (𝐻:𝑃onto→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ran 𝐻 = (𝐹𝐴)))
504, 48, 49sylanbrc 581 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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  46374
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