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Theorem foco2 6522
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 6521 . . . . . 6 (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧))
2 ffvelrn 6500 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
3 fvco3 6417 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
4 fveq2 6332 . . . . . . . . . . 11 (𝑥 = (𝐺𝑧) → (𝐹𝑥) = (𝐹‘(𝐺𝑧)))
54eqeq2d 2781 . . . . . . . . . 10 (𝑥 = (𝐺𝑧) → (((𝐹𝐺)‘𝑧) = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))))
65rspcev 3460 . . . . . . . . 9 (((𝐺𝑧) ∈ 𝐵 ∧ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
72, 3, 6syl2anc 573 . . . . . . . 8 ((𝐺:𝐴𝐵𝑧𝐴) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
8 eqeq1 2775 . . . . . . . . 9 (𝑦 = ((𝐹𝐺)‘𝑧) → (𝑦 = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
98rexbidv 3200 . . . . . . . 8 (𝑦 = ((𝐹𝐺)‘𝑧) → (∃𝑥𝐵 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
107, 9syl5ibrcom 237 . . . . . . 7 ((𝐺:𝐴𝐵𝑧𝐴) → (𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1110rexlimdva 3179 . . . . . 6 (𝐺:𝐴𝐵 → (∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
121, 11syl5 34 . . . . 5 (𝐺:𝐴𝐵 → (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1312impl 443 . . . 4 (((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
1413ralrimiva 3115 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
1514anim2i 603 . 2 ((𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)) → (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
16 3anass 1080 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ↔ (𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)))
17 dffo3 6517 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
1815, 16, 173imtr4i 281 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ccom 5253  wf 6027  ontowfo 6029  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039
This theorem is referenced by: (None)
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