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Theorem imasetpreimafvbijlemf1 46062
Description: Lemma for imasetpreimafvbij 46064: 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 46059 . 2 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
41, 2imasetpreimafvbijlemfv1 46061 . . . . 5 ((𝐹 Fn 𝐴𝑠𝑃) → ∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏))
51, 2imasetpreimafvbijlemfv1 46061 . . . . 5 ((𝐹 Fn 𝐴𝑟𝑃) → ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎))
64, 5anim12dan 619 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) ∧ ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎)))
7 eqeq12 2749 . . . . . . . . . . . 12 (((𝐻𝑠) = (𝐹𝑏) ∧ (𝐻𝑟) = (𝐹𝑎)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
87ancoms 459 . . . . . . . . . . 11 (((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
98adantl 482 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
10 simplll 773 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → 𝐹 Fn 𝐴)
11 simpllr 774 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑠𝑃𝑟𝑃))
12 simpr 485 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → 𝑏𝑠)
1312anim1i 615 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑏𝑠𝑎𝑟))
141elsetpreimafveq 46055 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃) ∧ (𝑏𝑠𝑎𝑟)) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1510, 11, 13, 14syl3anc 1371 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1615adantr 481 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
179, 16sylbid 239 . . . . . . . . 9 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
1817exp32 421 . . . . . . . 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 411 . . . 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 3200 . 2 (𝐹 Fn 𝐴 → ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
25 dff13 7253 . 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 205  wa 396   = wceq 1541  wcel 2106  {cab 2709  wral 3061  wrex 3070  {csn 4628   cuni 4908  cmpt 5231  ccnv 5675  cima 5679   Fn wfn 6538  wf 6539  1-1wf1 6540  cfv 6543
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fv 6551
This theorem is referenced by:  imasetpreimafvbij  46064
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