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Theorem foco2 6645
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 6644 . . . . . 6 (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧))
2 ffvelrn 6623 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → (𝐺𝑧) ∈ 𝐵)
3 fvco3 6537 . . . . . . . . 9 ((𝐺:𝐴𝐵𝑧𝐴) → ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧)))
4 fveq2 6448 . . . . . . . . . 10 (𝑥 = (𝐺𝑧) → (𝐹𝑥) = (𝐹‘(𝐺𝑧)))
54rspceeqv 3529 . . . . . . . . 9 (((𝐺𝑧) ∈ 𝐵 ∧ ((𝐹𝐺)‘𝑧) = (𝐹‘(𝐺𝑧))) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
62, 3, 5syl2anc 579 . . . . . . . 8 ((𝐺:𝐴𝐵𝑧𝐴) → ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥))
7 eqeq1 2782 . . . . . . . . 9 (𝑦 = ((𝐹𝐺)‘𝑧) → (𝑦 = (𝐹𝑥) ↔ ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
87rexbidv 3237 . . . . . . . 8 (𝑦 = ((𝐹𝐺)‘𝑧) → (∃𝑥𝐵 𝑦 = (𝐹𝑥) ↔ ∃𝑥𝐵 ((𝐹𝐺)‘𝑧) = (𝐹𝑥)))
96, 8syl5ibrcom 239 . . . . . . 7 ((𝐺:𝐴𝐵𝑧𝐴) → (𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
109rexlimdva 3213 . . . . . 6 (𝐺:𝐴𝐵 → (∃𝑧𝐴 𝑦 = ((𝐹𝐺)‘𝑧) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
111, 10syl5 34 . . . . 5 (𝐺:𝐴𝐵 → (((𝐹𝐺):𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥)))
1211impl 449 . . . 4 (((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ∧ 𝑦𝐶) → ∃𝑥𝐵 𝑦 = (𝐹𝑥))
1312ralrimiva 3148 . . 3 ((𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥))
1413anim2i 610 . 2 ((𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)) → (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
15 3anass 1079 . 2 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) ↔ (𝐹:𝐵𝐶 ∧ (𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶)))
16 dffo3 6640 . 2 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ∀𝑦𝐶𝑥𝐵 𝑦 = (𝐹𝑥)))
1714, 15, 163imtr4i 284 1 ((𝐹:𝐵𝐶𝐺:𝐴𝐵 ∧ (𝐹𝐺):𝐴onto𝐶) → 𝐹:𝐵onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  wrex 3091  ccom 5361  wf 6133  ontowfo 6135  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fo 6143  df-fv 6145
This theorem is referenced by: (None)
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