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Mirrors > Home > MPE Home > Th. List > Mathboxes > focofob | Structured version Visualization version GIF version |
Description: If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 47029 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.) |
Ref | Expression |
---|---|
focofob | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6737 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fnfocofob 47029 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷)) | |
3 | 1, 2 | syl3an1 1162 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ 𝐺:𝐶–onto→𝐷)) |
4 | dffn4 6827 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
5 | 1, 4 | sylib 218 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹) |
6 | 5 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴–onto→ran 𝐹) |
7 | foeq3 6819 | . . . . 5 ⊢ (ran 𝐹 = 𝐶 → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐶)) | |
8 | 7 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–onto→ran 𝐹 ↔ 𝐹:𝐴–onto→𝐶)) |
9 | 6, 8 | mpbid 232 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴–onto→𝐶) |
10 | 9 | biantrurd 532 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐺:𝐶–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷))) |
11 | 3, 10 | bitrd 279 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ran crn 5690 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 –onto→wfo 6561 |
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-sbc 3792 df-csb 3909 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: (None) |
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