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Theorem fcoresfo 45395
Description: If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-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)
fcoresfo.s (𝜑 → (𝐺𝐹):𝑃onto𝐷)
Assertion
Ref Expression
fcoresfo (𝜑𝑌:𝐸onto𝐷)

Proof of Theorem fcoresfo
StepHypRef Expression
1 fcores.g . . . 4 (𝜑𝐺:𝐶𝐷)
2 fcores.e . . . . . 6 𝐸 = (ran 𝐹𝐶)
32a1i 11 . . . . 5 (𝜑𝐸 = (ran 𝐹𝐶))
4 inss2 4193 . . . . 5 (ran 𝐹𝐶) ⊆ 𝐶
53, 4eqsstrdi 4002 . . . 4 (𝜑𝐸𝐶)
61, 5fssresd 6713 . . 3 (𝜑 → (𝐺𝐸):𝐸𝐷)
7 fcores.y . . . 4 𝑌 = (𝐺𝐸)
87feq1i 6663 . . 3 (𝑌:𝐸𝐷 ↔ (𝐺𝐸):𝐸𝐷)
96, 8sylibr 233 . 2 (𝜑𝑌:𝐸𝐷)
10 fcores.f . . . 4 (𝜑𝐹:𝐴𝐵)
11 fcores.p . . . 4 𝑃 = (𝐹𝐶)
12 fcores.x . . . 4 𝑋 = (𝐹𝑃)
1310, 2, 11, 12fcoreslem3 45389 . . 3 (𝜑𝑋:𝑃onto𝐸)
14 fof 6760 . . 3 (𝑋:𝑃onto𝐸𝑋:𝑃𝐸)
1513, 14syl 17 . 2 (𝜑𝑋:𝑃𝐸)
16 fcoresfo.s . . 3 (𝜑 → (𝐺𝐹):𝑃onto𝐷)
1710, 2, 11, 12, 1, 7fcores 45391 . . . . 5 (𝜑 → (𝐺𝐹) = (𝑌𝑋))
1817eqcomd 2739 . . . 4 (𝜑 → (𝑌𝑋) = (𝐺𝐹))
19 foeq1 6756 . . . 4 ((𝑌𝑋) = (𝐺𝐹) → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2018, 19syl 17 . . 3 (𝜑 → ((𝑌𝑋):𝑃onto𝐷 ↔ (𝐺𝐹):𝑃onto𝐷))
2116, 20mpbird 257 . 2 (𝜑 → (𝑌𝑋):𝑃onto𝐷)
22 foco2 7061 . 2 ((𝑌:𝐸𝐷𝑋:𝑃𝐸 ∧ (𝑌𝑋):𝑃onto𝐷) → 𝑌:𝐸onto𝐷)
239, 15, 21, 22syl3anc 1372 1 (𝜑𝑌:𝐸onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  cin 3913  ccnv 5636  ran crn 5638  cres 5639  cima 5640  ccom 5641  wf 6496  ontowfo 6498
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-sep 5260  ax-nul 5267  ax-pr 5388
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-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-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-fo 6506  df-fv 6508
This theorem is referenced by:  fcoresfob  45396  funfocofob  45400
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