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Theorem fcoresf1b 46515
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 479 . . . 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 479 . . . 4 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → 𝐺:𝐶𝐷)
8 fcores.y . . . 4 𝑌 = (𝐺𝐸)
9 simpr 483 . . . 4 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → (𝐺𝐹):𝑃1-1𝐷)
102, 3, 4, 5, 7, 8, 9fcoresf1 46514 . . 3 ((𝜑 ∧ (𝐺𝐹):𝑃1-1𝐷) → (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷))
1110ex 411 . 2 (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 → (𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷)))
12 f1co 6800 . . . 4 ((𝑌:𝐸1-1𝐷𝑋:𝑃1-1𝐸) → (𝑌𝑋):𝑃1-1𝐷)
1312ancoms 457 . . 3 ((𝑋:𝑃1-1𝐸𝑌:𝐸1-1𝐷) → (𝑌𝑋):𝑃1-1𝐷)
141, 3, 4, 5, 6, 8fcores 46512 . . . 4 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
15 f1eq1 6783 . . . 4 ((𝐺𝐹) = (𝑌𝑋) → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑌𝑋):𝑃1-1𝐷))
1614, 15syl 17 . . 3 (𝜑 → ((𝐺𝐹):𝑃1-1𝐷 ↔ (𝑌𝑋):𝑃1-1𝐷))
1713, 16imbitrrid 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 394   = wceq 1533  cin 3938  ccnv 5671  ran crn 5673  cres 5674  cima 5675  ccom 5676  wf 6539  1-1wf1 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-fv 6551
This theorem is referenced by:  fcoresf1ob  46518
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