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Theorem imasetpreimafvbijlemf1 47391
Description: Lemma for imasetpreimafvbij 47393: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.h 𝐻 = (𝑝𝑃 (𝐹𝑝))
Assertion
Ref Expression
imasetpreimafvbijlemf1 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃
Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemf1
Dummy variables 𝑎 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fundcmpsurinj.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
2 fundcmpsurinj.h . . 3 𝐻 = (𝑝𝑃 (𝐹𝑝))
31, 2imasetpreimafvbijlemf 47388 . 2 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
41, 2imasetpreimafvbijlemfv1 47390 . . . . 5 ((𝐹 Fn 𝐴𝑠𝑃) → ∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏))
51, 2imasetpreimafvbijlemfv1 47390 . . . . 5 ((𝐹 Fn 𝐴𝑟𝑃) → ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎))
64, 5anim12dan 619 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) ∧ ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎)))
7 eqeq12 2754 . . . . . . . . . . . 12 (((𝐻𝑠) = (𝐹𝑏) ∧ (𝐻𝑟) = (𝐹𝑎)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
87ancoms 458 . . . . . . . . . . 11 (((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
98adantl 481 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
10 simplll 775 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → 𝐹 Fn 𝐴)
11 simpllr 776 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑠𝑃𝑟𝑃))
12 simpr 484 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → 𝑏𝑠)
1312anim1i 615 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑏𝑠𝑎𝑟))
141elsetpreimafveq 47384 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃) ∧ (𝑏𝑠𝑎𝑟)) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1510, 11, 13, 14syl3anc 1373 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1615adantr 480 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
179, 16sylbid 240 . . . . . . . . 9 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
1817exp32 420 . . . . . . . 8 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐹𝑏) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
1918rexlimdva 3155 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → (∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐹𝑏) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
2019com23 86 . . . . . 6 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → ((𝐻𝑠) = (𝐹𝑏) → (∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
2120rexlimdva 3155 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) → (∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
2221impd 410 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → ((∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) ∧ ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎)) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟)))
236, 22mpd 15 . . 3 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
2423ralrimivva 3202 . 2 (𝐹 Fn 𝐴 → ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
25 dff13 7275 . 2 (𝐻:𝑃1-1→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟)))
263, 24, 25sylanbrc 583 1 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {csn 4626   cuni 4907  cmpt 5225  ccnv 5684  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  imasetpreimafvbij  47393
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