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Theorem imasetpreimafvbijlemf1 45686
Description: Lemma for imasetpreimafvbij 45688: 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 45683 . 2 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
41, 2imasetpreimafvbijlemfv1 45685 . . . . 5 ((𝐹 Fn 𝐴𝑠𝑃) → ∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏))
51, 2imasetpreimafvbijlemfv1 45685 . . . . 5 ((𝐹 Fn 𝐴𝑟𝑃) → ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎))
64, 5anim12dan 620 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) ∧ ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎)))
7 eqeq12 2750 . . . . . . . . . . . 12 (((𝐻𝑠) = (𝐹𝑏) ∧ (𝐻𝑟) = (𝐹𝑎)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
87ancoms 460 . . . . . . . . . . 11 (((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
98adantl 483 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
10 simplll 774 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → 𝐹 Fn 𝐴)
11 simpllr 775 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑠𝑃𝑟𝑃))
12 simpr 486 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → 𝑏𝑠)
1312anim1i 616 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑏𝑠𝑎𝑟))
141elsetpreimafveq 45679 . . . . . . . . . . . 12 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃) ∧ (𝑏𝑠𝑎𝑟)) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1510, 11, 13, 14syl3anc 1372 . . . . . . . . . . 11 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
1615adantr 482 . . . . . . . . . 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 422 . . . . . . . 8 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐹𝑏) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
1918rexlimdva 3149 . . . . . . 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 3149 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) → (∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
2221impd 412 . . . 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 3194 . 2 (𝐹 Fn 𝐴 → ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
25 dff13 7206 . 2 (𝐻:𝑃1-1→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟)))
263, 24, 25sylanbrc 584 1 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  {csn 4590   cuni 4869  cmpt 5192  ccnv 5636  cima 5640   Fn wfn 6495  wf 6496  1-1wf1 6497  cfv 6500
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508
This theorem is referenced by:  imasetpreimafvbij  45688
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