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Mirrors > Home > MPE Home > Th. List > focofo | Structured version Visualization version GIF version |
Description: Composition of onto functions. Generalisation of foco 6771. (Contributed by AV, 29-Sep-2024.) |
Ref | Expression |
---|---|
focofo | ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fof 6757 | . . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | fcof 6692 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵) | |
3 | 1, 2 | sylan 581 | . . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵) |
4 | 3 | 3adant3 1133 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵) |
5 | rnco 6205 | . . 3 ⊢ ran (𝐹 ∘ 𝐺) = ran (𝐹 ↾ ran 𝐺) | |
6 | 1 | freld 6675 | . . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Rel 𝐹) |
7 | 6 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → Rel 𝐹) |
8 | fdm 6678 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
9 | 8 | eqcomd 2743 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹) |
10 | 1, 9 | syl 17 | . . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐴 = dom 𝐹) |
11 | 10 | sseq1d 3976 | . . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐴 ⊆ ran 𝐺 ↔ dom 𝐹 ⊆ ran 𝐺)) |
12 | 11 | biimpa 478 | . . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐴 ⊆ ran 𝐺) → dom 𝐹 ⊆ ran 𝐺) |
13 | relssres 5979 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → (𝐹 ↾ ran 𝐺) = 𝐹) | |
14 | 13 | rneqd 5894 | . . . . 5 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹) |
15 | 7, 12, 14 | 3imp3i2an 1346 | . . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = ran 𝐹) |
16 | forn 6760 | . . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
17 | 16 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran 𝐹 = 𝐵) |
18 | 15, 17 | eqtrd 2777 | . . 3 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ↾ ran 𝐺) = 𝐵) |
19 | 5, 18 | eqtrid 2789 | . 2 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → ran (𝐹 ∘ 𝐺) = 𝐵) |
20 | dffo2 6761 | . 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵 ∧ ran (𝐹 ∘ 𝐺) = 𝐵)) | |
21 | 4, 19, 20 | sylanbrc 584 | 1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊆ wss 3911 ◡ccnv 5633 dom cdm 5634 ran crn 5635 ↾ cres 5636 “ cima 5637 ∘ ccom 5638 Rel wrel 5639 Fun wfun 6491 ⟶wf 6493 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: foco 6771 funfocofob 45317 |
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