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Theorem foco2 6985
Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
Assertion
Ref Expression
foco2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)

Proof of Theorem foco2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 foelrn 6984 . . . . . 6 (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧))
2 ffvelrn 6961 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
3 fvco3 6869 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
4 fveq2 6776 . . . . . . . . . 10 (𝑥 = (𝐺𝑧) → (𝐹𝑥) = (𝐹‘(𝐺𝑧)))
54rspceeqv 3576 . . . . . . . . 9 (((𝐺𝑧) ∈ 𝐵 ∧ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
62, 3, 5syl2anc 584 . . . . . . . 8 ((𝐺:𝐴𝐵𝑧𝐴) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
7 eqeq1 2742 . . . . . . . . 9 (𝑦 = ((𝐹𝐺)‘𝑧) → (𝑦 = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
87rexbidv 3225 . . . . . . . 8 (𝑦 = ((𝐹𝐺)‘𝑧) → (∃𝑥𝐵 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
96, 8syl5ibrcom 246 . . . . . . 7 ((𝐺:𝐴𝐵𝑧𝐴) → (𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
109rexlimdva 3212 . . . . . 6 (𝐺:𝐴𝐵 → (∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
111, 10syl5 34 . . . . 5 (𝐺:𝐴𝐵 → (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1211impl 456 . . . 4 (((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
1312ralrimiva 3103 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
1413anim2i 617 . 2 ((𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)) → (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
15 3anass 1094 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ↔ (𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)))
16 dffo3 6980 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
1714, 15, 163imtr4i 292 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  ccom 5595  wf 6431  ontowfo 6433  cfv 6435
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 5225  ax-nul 5232  ax-pr 5354
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 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-fo 6441  df-fv 6443
This theorem is referenced by:  fcoresfo  44532
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