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Theorem imasetpreimafvbijlemf1 44390
Description: Lemma for imasetpreimafvbij 44392: 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 44387 . 2 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
41, 2imasetpreimafvbijlemfv1 44389 . . . . 5 ((𝐹 Fn 𝐴𝑠𝑃) → ∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏))
51, 2imasetpreimafvbijlemfv1 44389 . . . . 5 ((𝐹 Fn 𝐴𝑟𝑃) → ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎))
64, 5anim12dan 622 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) ∧ ∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎)))
7 eqeq12 2752 . . . . . . . . . . . 12 (((𝐻𝑠) = (𝐹𝑏) ∧ (𝐻𝑟) = (𝐹𝑎)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
87ancoms 462 . . . . . . . . . . 11 (((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏)) → ((𝐻𝑠) = (𝐻𝑟) ↔ (𝐹𝑏) = (𝐹𝑎)))
98adantl 485 . . . . . . . . . 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 488 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) → 𝑏𝑠)
1312anim1i 618 . . . . . . . . . . . 12 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → (𝑏𝑠𝑎𝑟))
141elsetpreimafveq 44383 . . . . . . . . . . . 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 484 . . . . . . . . . 10 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐹𝑏) = (𝐹𝑎) → 𝑠 = 𝑟))
179, 16sylbid 243 . . . . . . . . 9 (((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) ∧ ((𝐻𝑟) = (𝐹𝑎) ∧ (𝐻𝑠) = (𝐹𝑏))) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
1817exp32 424 . . . . . . . 8 ((((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) ∧ 𝑏𝑠) ∧ 𝑎𝑟) → ((𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐹𝑏) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
1918rexlimdva 3194 . . . . . . 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 3194 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑠𝑃𝑟𝑃)) → (∃𝑏𝑠 (𝐻𝑠) = (𝐹𝑏) → (∃𝑎𝑟 (𝐻𝑟) = (𝐹𝑎) → ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))))
2221impd 414 . . . 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 3103 . 2 (𝐹 Fn 𝐴 → ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟))
25 dff13 7024 . 2 (𝐻:𝑃1-1→(𝐹𝐴) ↔ (𝐻:𝑃⟶(𝐹𝐴) ∧ ∀𝑠𝑃𝑟𝑃 ((𝐻𝑠) = (𝐻𝑟) → 𝑠 = 𝑟)))
263, 24, 25sylanbrc 586 1 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wrex 3054  {csn 4516   cuni 4796  cmpt 5110  ccnv 5524  cima 5528   Fn wfn 6334  wf 6335  1-1wf1 6336  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fv 6347
This theorem is referenced by:  imasetpreimafvbij  44392
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