Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fcoresf1b Structured version   Visualization version   GIF version

Theorem fcoresf1b 44542
Description: A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.)
Hypotheses
Ref Expression
fcores.f (𝜑𝐹:𝐴𝐵)
fcores.e 𝐸 = (ran 𝐹𝐶)
fcores.p 𝑃 = (𝐹𝐶)
fcores.x 𝑋 = (𝐹𝑃)
fcores.g (𝜑𝐺:𝐶𝐷)
fcores.y 𝑌 = (𝐺𝐸)
Assertion
Ref Expression
fcoresf1b (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷)))

Proof of Theorem fcoresf1b
StepHypRef Expression
1 fcores.f . . . . 5 (𝜑𝐹:𝐴𝐵)
21adantr 481 . . . 4 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → 𝐹:𝐴𝐵)
3 fcores.e . . . 4 𝐸 = (ran 𝐹𝐶)
4 fcores.p . . . 4 𝑃 = (𝐹𝐶)
5 fcores.x . . . 4 𝑋 = (𝐹𝑃)
6 fcores.g . . . . 5 (𝜑𝐺:𝐶𝐷)
76adantr 481 . . . 4 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → 𝐺:𝐶𝐷)
8 fcores.y . . . 4 𝑌 = (𝐺𝐸)
9 simpr 485 . . . 4 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → (𝐺𝐹):𝑃1-1𝐷)
102, 3, 4, 5, 7, 8, 9fcoresf1 44541 . . 3 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷))
1110ex 413 . 2 (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 → (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷)))
12 f1co 6674 . . . 4 ((𝑌:𝐸1-1𝐷𝑋:𝑃1-1𝐸) → (𝑌𝑋):𝑃1-1𝐷)
1312ancoms 459 . . 3 ((𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷) → (𝑌𝑋):𝑃1-1𝐷)
141, 3, 4, 5, 6, 8fcores 44539 . . . 4 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
15 f1eq1 6657 . . . 4 ((𝐺𝐹) = (𝑌𝑋) → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑌𝑋):𝑃1-1𝐷))
1614, 15syl 17 . . 3 (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑌𝑋):𝑃1-1𝐷))
1713, 16syl5ibr 245 . 2 (𝜑 → ((𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷) → (𝐺𝐹):𝑃1-1𝐷))
1811, 17impbid 211 1 (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  cin 3885  ccnv 5583  ran crn 5585  cres 5586  cima 5587  ccom 5588  wf 6422  1-1wf1 6423
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-fv 6434
This theorem is referenced by:  fcoresf1ob  44545
  Copyright terms: Public domain W3C validator