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Theorem foco2 6335
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 6334 . . . . . 6 (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧))
2 ffvelrn 6313 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
3 fvco3 6232 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
4 fveq2 6148 . . . . . . . . . . 11 (𝑥 = (𝐺𝑧) → (𝐹𝑥) = (𝐹‘(𝐺𝑧)))
54eqeq2d 2631 . . . . . . . . . 10 (𝑥 = (𝐺𝑧) → (((𝐹𝐺)‘𝑧) = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))))
65rspcev 3295 . . . . . . . . 9 (((𝐺𝑧) ∈ 𝐵 ∧ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
72, 3, 6syl2anc 692 . . . . . . . 8 ((𝐺:𝐴𝐵𝑧𝐴) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
8 eqeq1 2625 . . . . . . . . 9 (𝑦 = ((𝐹𝐺)‘𝑧) → (𝑦 = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
98rexbidv 3045 . . . . . . . 8 (𝑦 = ((𝐹𝐺)‘𝑧) → (∃𝑥𝐵 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
107, 9syl5ibrcom 237 . . . . . . 7 ((𝐺:𝐴𝐵𝑧𝐴) → (𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1110rexlimdva 3024 . . . . . 6 (𝐺:𝐴𝐵 → (∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
121, 11syl5 34 . . . . 5 (𝐺:𝐴𝐵 → (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1312impl 649 . . . 4 (((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
1413ralrimiva 2960 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
1514anim2i 592 . 2 ((𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)) → (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
16 3anass 1040 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ↔ (𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)))
17 dffo3 6330 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ccom 5078  wf 5843  ontowfo 5845  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855
This theorem is referenced by: (None)
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