Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3f1oss2 Structured version   Visualization version   GIF version

Theorem 3f1oss2 46976
Description: The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
Assertion
Ref Expression
3f1oss2 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐵𝐷𝐼)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))

Proof of Theorem 3f1oss2
StepHypRef Expression
1 f1ocnv 6855 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
2 id 22 . . 3 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶1-1-onto𝐷)
3 f1ocnv 6855 . . 3 (𝐻:𝐸1-1-onto𝐼𝐻:𝐼1-1-onto𝐸)
4 3f1oss1 46975 . . 3 (((𝐹:𝐵1-1-onto𝐴𝐺:𝐶1-1-onto𝐷𝐻:𝐼1-1-onto𝐸) ∧ (𝐶𝐵𝐷𝐼)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
51, 2, 3, 4syl3anl 1413 . 2 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐵𝐷𝐼)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
6 f1orel 6846 . . . . 5 (𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)
7 dfrel2 6205 . . . . . . . . 9 (Rel 𝐹𝐹 = 𝐹)
87biimpi 216 . . . . . . . 8 (Rel 𝐹𝐹 = 𝐹)
98eqcomd 2739 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
109coeq2d 5870 . . . . . 6 (Rel 𝐹 → ((𝐻𝐺) ∘ 𝐹) = ((𝐻𝐺) ∘ 𝐹))
1110f1oeq1d 6838 . . . . 5 (Rel 𝐹 → (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷)))
126, 11syl 17 . . . 4 (𝐹:𝐴1-1-onto𝐵 → (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷)))
13123ad2ant1 1131 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) → (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷)))
1413adantr 480 . 2 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐵𝐷𝐼)) → (((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷) ↔ ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷)))
155, 14mpbird 257 1 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐻:𝐸1-1-onto𝐼) ∧ (𝐶𝐵𝐷𝐼)) → ((𝐻𝐺) ∘ 𝐹):(𝐹𝐶)–1-1-onto→(𝐻𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1085   = wceq 1535  wss 3963  ccnv 5682  cima 5686  ccom 5687  Rel wrel 5688  1-1-ontowf1o 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566
This theorem is referenced by:  uspgrlim  47817
  Copyright terms: Public domain W3C validator