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Mirrors > Home > MPE Home > Th. List > fcofo | Structured version Visualization version GIF version |
Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
fcofo | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴⟶𝐵) | |
2 | ffvelcdm 7101 | . . . . 5 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴) | |
3 | 2 | 3ad2antl2 1185 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝑆‘𝑦) ∈ 𝐴) |
4 | simpl3 1192 | . . . . . 6 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) | |
5 | 4 | fveq1d 6909 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (( I ↾ 𝐵)‘𝑦)) |
6 | fvco3 7008 | . . . . . 6 ⊢ ((𝑆:𝐵⟶𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦))) | |
7 | 6 | 3ad2antl2 1185 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ((𝐹 ∘ 𝑆)‘𝑦) = (𝐹‘(𝑆‘𝑦))) |
8 | fvresi 7193 | . . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦) | |
9 | 8 | adantl 481 | . . . . 5 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦) |
10 | 5, 7, 9 | 3eqtr3rd 2784 | . . . 4 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑆‘𝑦))) |
11 | fveq2 6907 | . . . . 5 ⊢ (𝑥 = (𝑆‘𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑆‘𝑦))) | |
12 | 11 | rspceeqv 3645 | . . . 4 ⊢ (((𝑆‘𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹‘(𝑆‘𝑦))) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
13 | 3, 10, 12 | syl2anc 584 | . . 3 ⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
14 | 13 | ralrimiva 3144 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)) |
15 | dffo3 7122 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) | |
16 | 1, 14, 15 | sylanbrc 583 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 I cid 5582 ↾ cres 5691 ∘ ccom 5693 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 |
This theorem is referenced by: fcof1od 7314 smndex2dnrinv 18941 |
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