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Theorem imasetpreimafvbijlemfo 45587
Description: Lemma for imasetpreimafvbij 45588: 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 45583 . . 3 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
43adantr 481 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃⟶(𝐹𝐴))
51preimafvelsetpreimafv 45570 . . . . . . . . 9 ((𝐹 Fn 𝐴𝐴𝑉𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
653expa 1118 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ {(𝐹𝑎)}) ∈ 𝑃)
7 imaeq2 6009 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
87unieqd 4879 . . . . . . . . . 10 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑝) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
98eqeq2d 2747 . . . . . . . . 9 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
109adantl 482 . . . . . . . 8 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → ((𝐹𝑎) = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)}))))
11 uniimaprimaeqfv 45564 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1211adantlr 713 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑎))
1312eqcomd 2742 . . . . . . . 8 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝐹𝑎) = (𝐹 “ (𝐹 “ {(𝐹𝑎)})))
146, 10, 13rspcedvd 3583 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝))
15 eqeq1 2740 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1615eqcoms 2744 . . . . . . . 8 ((𝐹𝑎) = 𝑦 → (𝑦 = (𝐹𝑝) ↔ (𝐹𝑎) = (𝐹𝑝)))
1716rexbidv 3175 . . . . . . 7 ((𝐹𝑎) = 𝑦 → (∃𝑝𝑃 𝑦 = (𝐹𝑝) ↔ ∃𝑝𝑃 (𝐹𝑎) = (𝐹𝑝)))
1814, 17syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → ((𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
1918rexlimdva 3152 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 → ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
208eqcomd 2742 . . . . . . . . . . 11 (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹 “ (𝐹 “ {(𝐹𝑎)})) = (𝐹𝑝))
2113, 20sylan9eq 2796 . . . . . . . . . 10 ((((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) ∧ 𝑝 = (𝐹 “ {(𝐹𝑎)})) → (𝐹𝑎) = (𝐹𝑝))
2221ex 413 . . . . . . . . 9 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑎𝐴) → (𝑝 = (𝐹 “ {(𝐹𝑎)}) → (𝐹𝑎) = (𝐹𝑝)))
2322reximdva 3165 . . . . . . . 8 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
241elsetpreimafv 45567 . . . . . . . . 9 (𝑝𝑃 → ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
25 fveq2 6842 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝐹𝑎) = (𝐹𝑥))
2625sneqd 4598 . . . . . . . . . . . 12 (𝑎 = 𝑥 → {(𝐹𝑎)} = {(𝐹𝑥)})
2726imaeq2d 6013 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝐹 “ {(𝐹𝑎)}) = (𝐹 “ {(𝐹𝑥)}))
2827eqeq2d 2747 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ 𝑝 = (𝐹 “ {(𝐹𝑥)})))
2928cbvrexvw 3226 . . . . . . . . 9 (∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}) ↔ ∃𝑥𝐴 𝑝 = (𝐹 “ {(𝐹𝑥)}))
3024, 29sylibr 233 . . . . . . . 8 (𝑝𝑃 → ∃𝑎𝐴 𝑝 = (𝐹 “ {(𝐹𝑎)}))
3123, 30impel 506 . . . . . . 7 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝))
32 eqeq2 2748 . . . . . . . 8 (𝑦 = (𝐹𝑝) → ((𝐹𝑎) = 𝑦 ↔ (𝐹𝑎) = (𝐹𝑝)))
3332rexbidv 3175 . . . . . . 7 (𝑦 = (𝐹𝑝) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑎𝐴 (𝐹𝑎) = (𝐹𝑝)))
3431, 33syl5ibrcom 246 . . . . . 6 (((𝐹 Fn 𝐴𝐴𝑉) ∧ 𝑝𝑃) → (𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3534rexlimdva 3152 . . . . 5 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑝𝑃 𝑦 = (𝐹𝑝) → ∃𝑎𝐴 (𝐹𝑎) = 𝑦))
3619, 35impbid 211 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (∃𝑎𝐴 (𝐹𝑎) = 𝑦 ↔ ∃𝑝𝑃 𝑦 = (𝐹𝑝)))
3736abbidv 2805 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦} = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
38 fnfun 6602 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
39 fndm 6605 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
40 eqimss2 4001 . . . . . . 7 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
4139, 40syl 17 . . . . . 6 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
4238, 41jca 512 . . . . 5 (𝐹 Fn 𝐴 → (Fun 𝐹𝐴 ⊆ dom 𝐹))
4342adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → (Fun 𝐹𝐴 ⊆ dom 𝐹))
44 dfimafn 6905 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
4543, 44syl 17 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → (𝐹𝐴) = {𝑦 ∣ ∃𝑎𝐴 (𝐹𝑎) = 𝑦})
462rnmpt 5910 . . . 4 ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)}
4746a1i 11 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = {𝑦 ∣ ∃𝑝𝑃 𝑦 = (𝐹𝑝)})
4837, 45, 473eqtr4rd 2787 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → ran 𝐻 = (𝐹𝐴))
49 dffo2 6760 . 2 (𝐻:𝑃onto→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ran 𝐻 = (𝐹𝐴)))
504, 48, 49sylanbrc 583 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  wss 3910  {csn 4586   cuni 4865  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by:  imasetpreimafvbij  45588
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